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+{"text":"\\section{Introduction}\n\n$Q$-balls represent stationary localized solutions \nof a complex scalar field theory with\na suitable self-interaction in flat space\n\\cite{Friedberg:1976me,Coleman:1985ki}.\nThe global phase invariance of the scalar field theory\nis associated with a conserved charge $Q$ \\cite{Friedberg:1976me},\nwhich represents the electromagnetic charge,\nonce the theory is promoted to a gauge theory.\n\nThe simplest type of $Q$-balls is spherically symmetric.\nThese possess a finite mass and charge, but carry no angular momentum.\nConsidering their mass as a function of their charge,\nthere are two branches of $Q$-balls, merging and ending at a cusp,\nwhere mass and charge assume their minimal values \\cite{Friedberg:1976me}.\n\nRecently, a new type of scalar potential for $Q$-balls was considered,\nleading to the signum-Gordan equation for the scalar field\n\\cite{Arodz:2008jk,Arodz:2008nm}.\nThis potential gives rise to spatially compact $Q$-balls, \nwhere the scalar field vanishes identically outside a critical\nradius $r_o$ \\cite{Arodz:2008jk}.\nWhen coupled to electromagnetism, a new type of solution appears,\n$Q$-shells \\cite{Arodz:2008nm}.\nIn $Q$-shells the scalar field vanishes identically both inside\na critical radius $r_i$ and outside a critical radius $r_o$,\nthus forming a finite shell $r_i < r < r_o$ of charged matter.\n\nWhen gravity is coupled to $Q$-balls, boson stars arise,\nrepresenting globally regular self-gravitating solutions\n\\cite{Lee:1991ax,Jetzer:1991jr,Mielke:2000mh,Schunck:2003kk}.\nThe presence of gravity has a crucial influence\non the domain of existence of the classical solutions.\nInstead of only two branches of solutions joint at a single cusp, \nthe boson stars exhibit an intricate cusp structure,\nwhere mass and charge oscillate endlessly.\nFor black holes with scalar fields, on the other hand, \na number of theorems exist, which exclude their existence\nunder a large variety of conditions \n\\cite{Bekenstein:1971hc,Bekenstein:1995un,Mayo:1996mv}.\n\nHere we consider the effect of gravity on the $Q$-balls and\n$Q$-shells of the signum-Gordon model coupled to a Maxwell field.\nWe construct these charged boson stars and gravitating $Q$-shells\nand analyze their properties and their domains of existence.\nWe observe that at certain critical values of the mass and charge,\nthe space-times form a throat at the (outer) radius $r_o$, \nrendering the respective exterior space-time\nan exterior extremal Reissner-Nordstr\\\"om space-time.\n\nMoreover, we show that in this model\nthe black hole theorems can be elluded,\nthat forbid black holes with scalar hair.\nIndeed, the gravitating $Q$-shells can be endowed with a horizon $r_H$\nin the interior region $0< r_H < r_i$,\nwhere the scalar field vanishes and the gauge potential is constant.\nThis Schwarzschild-type black hole in the interior\nis surrounded by a shell of charged matter, $r_i<r<r_o$,\nleading to an exterior space-time $r_o<r<\\infty$ of\nReissner-Nordstr\\\"om type. \nWe analyze the global and horizon properties of these black holes \nwithin $Q$-shells, and present a mass relation.\nWhen a throat develops at the outer radius $r_o$,\nwhich renders the respective exterior space-time\nan exterior extremal Reissner-Nordstr\\\"om space-time,\nthe temperature of the horizon $r_H$ tends to zero.\n\nIn section 2 we recall the action and ans\\\"atze for the fields.\nWe present the gravitating $Q$-balls and $Q$-shells in section 3 and 4, \nrespectively, and the black holes within $Q$-shells in section 5.\nWe end with a conclusion and an outlook.\n\n\\section{Action}\n\nWe consider the action of a self-interacting complex scalar field\n$\\Phi$ coupled to a U(1) gauge field and to Einstein gravity\n\\begin{equation}\nS=\\int \\left[ \\frac{R}{16\\pi G}\n   - \\frac{1}{4} F^{\\mu\\nu} F_{\\mu\\nu}\n   -  \\left( D_\\mu \\Phi \\right)^* \\left( D^\\mu \\Phi \\right)\n - U( \\left| \\Phi \\right|) \n \\right] \\sqrt{-g} d^4x\n , \\label{action}\n\\end{equation}\nwith field strength tensor\n\\begin{equation}\nF_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu \n , \\label{action2}\n\\end{equation}\ncovariant derivative\n\\begin{equation}\nD_\\mu \\Phi = \\partial_\\mu \\Phi + i e A_\\mu \\Phi\n , \\label{action3}\n\\end{equation}\ncurvature scalar $R$, Newton's constant $G$, \ngauge coupling constant $e$,\nand the asterisk denotes complex conjugation.\nThe scalar potential $U$ is chosen as\n\\begin{equation}\nU(|\\Phi|) =  \\lambda  |\\Phi| \n . \\label{U} \\end{equation} \n\nVariation of the action with respect to the metric and the matter fields\nleads, respectively, to the Einstein equations\n\\begin{equation}\nG_{\\mu\\nu}= R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}R = 8\\pi G T_{\\mu\\nu}\n\\  \\label{ee} \\end{equation}\nwith stress-energy tensor\n\\begin{eqnarray}\nT_{\\mu\\nu} &=& g_{\\mu\\nu}{L}_M\n-2 \\frac{\\partial {L}_M}{\\partial g^{\\mu\\nu}}\n=\n    ( F_{\\mu\\alpha} F_{\\nu\\beta} g^{\\alpha\\beta}\n   -\\frac{1}{4} g_{\\mu\\nu} F_{\\alpha\\beta} F^{\\alpha\\beta})\n\\nonumber\\\\\n&-& \n   \\frac{1}{2} g_{\\mu\\nu} \\left(     (D_\\alpha \\Phi)^* (D_\\beta \\Phi)\n  + (D_\\beta \\Phi)^* (D_\\alpha \\Phi)    \\right) g^{\\alpha\\beta}\n  + (D_\\mu \\Phi)^* (D_\\nu \\Phi) + (D_\\nu \\Phi)^* (D_\\mu \\Phi)\n\\nonumber\\\\\n&-& \n    {\\lambda} g_{\\mu\\nu}  |\\Phi| \n , \\label{tmunu}\n\\end{eqnarray}\nand the matter field equations,\n\\begin{eqnarray}\n& & \\partial_\\mu \\left ( \\sqrt{-g} F^{\\mu\\nu} \\right) =\n   \\sqrt{-g} e \\Phi^* D^\\nu \\Phi \n\\label{feqA} \\end{eqnarray}\n\\begin{eqnarray}\n& &D_\\mu D^\\mu \\Phi = - \\frac{\\lambda}{2} \\frac{\\Phi}{|\\Phi|}\n . \\label{feqH} \\end{eqnarray}\n\nTo construct static spherically symmetric solutions\nwe employ Schwarz\\-schild-like coordinates and adopt\nthe spherically symmetric metric\n\\begin{equation}\nds^2=g_{\\mu\\nu}dx^\\mu dx^\\nu=\n  -A^2N dt^2 + N^{-1} dr^2 + r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)\n , \\end{equation}\nwith\n\\begin{equation}\nN=1-\\frac{2m(r)}{r}\n . \\end{equation}\n\nFor solutions with vanishing magnetic field\nthe Ansatz for the matter fields has the form\n\\begin{equation}\n \\Phi = \\phi(r) e^{i \\omega t}\n , \\label{phi} \\end{equation}\n\\begin{equation}\n A_\\mu d x^\\mu = A_0(r) dt\n . \\label{A_0} \\end{equation}\n \nFor notational simplicity, we \nintroduce new coupling constants \\cite{Arodz:2008nm}\n\\begin{equation}\n\\alpha^2 = a = 4\\pi G \\frac{\\beta^{1\/3}}{e^2},\n\\ \\ \\\n\\beta = \\frac{\\lambda e}{\\sqrt{2}} \n , \\label{constants} \\end{equation}\nand redefine the matter field functions,\n\\begin{equation}\nh(r) = \\sqrt{2}e \\phi(r) ,\n\\ \\ \\\nb(r) = \\omega + e A_0(r)\n . \\label{functions} \\end{equation}\nThe latter corresponds to\nperforming a gauge transformation to make the scalar field real\nand absorbing the frequency $\\omega$ of the scalar field into the gauge\ntransformed vector potential.\nNote, that the parameter $\\beta$ can be removed by rescaling \nand will therefore be set to one\n\\cite{Arodz:2008nm}.\nThus the only parameter left is the gravitational coupling $\\alpha$.\n\nLet us now specify the boundary conditions for the \nmetric and matter functions.\nFor the metric function $A$ we adopt\n\\begin{equation}\nA(r_o)=1\n\\ , \\end{equation}\nwhere $r_o$ is the outer radius,\nthus fixing the time coordinate. \nFor the mass function $m(r)$ we require for globally regular ball-like\nboson star solutions\n\\begin{equation}\nm(0)=0 \n, \\end{equation}\nfor globally regular shell-like solutions \n\\begin{equation}\nm(r_i)=0 \n, \\end{equation}\nwhere $r_i$ is the inner radius of the shell,\nand for black hole solutions\n\\begin{equation}\nm(r_H)= \\frac{r_H}{2}\n  \\end{equation}\nwhere $r_H < r_i$ denotes the horizon.\n\nTo be able to specify more than four boundary conditions \nfor the matter functions we introduce one or more\nauxiliary variables.\nFor globally regular boson star solutions we require\nat the origin and at the outer radius $r_o$\nthe conditions\n\\begin{equation}\nb'(0)=0 , \\ \\ \\ h'(0)=0 , \\ \\ \\ h(r_o)=0 , \\ \\ \\ h'(r_o)=0\n, \\end{equation}\nwhere the prime denotes differentiation with respect to $r$.\nIn order to choose also the value of $b(0)$ as a boundary condition,\nwe make the outer radius $r_o$ an auxiliary (constant) variable,\nand thus add the differential equation $r_o\\,'=0$,\nwithout imposing a boundary condition.\n\nFor globally regular shell solutions as well as for black holes\nwe require at the inner radius $r_i$ and at the outer radius $r_o$\nthe conditions\n\\begin{equation}\nb'(r_i)=0 , \\ \\ \\ h(r_i)=0 , \\ \\ \\ h'(r_i)=0 , \\ \\ \\ h(r_o)=0 , \\ \\ \\ h'(r_o)=0\n . \\end{equation}\nIn order to choose also the value of $b(r_i)$ as a boundary condition,\nwe now also make the ratio of inner and outer radius $r_i\/r_o$\nan auxiliary (constant) variable.\nAlternatively to demanding a certain value for $b(r_i)$,\nwe may also specify the value of the electric charge $Q$.\n\nThe charge $Q$ of the solutions is associated with the conserved current\n\\begin{eqnarray}\nj^{\\mu} & = &  - i \\left( \\Phi^* D^{\\mu} \\Phi\n - \\Phi D^{\\mu}\\Phi ^* \\right) \\ , \\ \\ \\\nj^{\\mu} _{\\ ; \\, \\mu}  =  0 \\ ,\n\\end{eqnarray}\ni.e., the charge is obtained as the spatial integral\n\\begin{equation}\nQ = -i \\int j^0 \\sqrt{-g} dr d\\theta d\\varphi\n. \\end{equation}\n\nThe mass $M$ of the stationary asymptotically flat space-times\nis obtained from the corresponding Komar expression.\nFor globally regular space-times like boson stars\nand shells of boson matter the mass is given by\n\\begin{equation}\nM = \n \\frac{1}{{4\\pi G}} \\int_{\\Sigma}\n R_{\\mu\\nu}n^\\mu\\xi^\\nu dV\n , \\label{komarM1}\n\\end{equation}\nwhere $\\Sigma$ denotes an asymptotically flat spacelike hypersurface,\n$n^\\mu$ is normal to $\\Sigma$ with $n_\\mu n^\\mu = -1$,\n$dV$ is the natural volume element on $\\Sigma$,\nand $\\xi$ denotes an asymptotically timelike Killing vector field\n\\cite{wald}.\nReplacing the Ricci tensor via the Einstein equations by the\nstress-energy tensor yields\n\\begin{equation}\nM = \n  \\, 2 \\int_{\\Sigma} \\left(  T_{\\mu \\nu}\n-\\frac{1}{2} \\, g_{\\mu\\nu} \\, T_{\\gamma}^{\\ \\gamma}\n \\right) n^{\\mu }\\xi^{\\nu} dV\n . \\label{komarM2}\n\\end{equation}\n\nFor black hole space-times the corresponding Komar expression is given by\n\\begin{equation}\nM = M_H +\n  \\, 2 \\int_{\\Sigma} \\left(  T_{\\mu \\nu}\n-\\frac{1}{2} \\, g_{\\mu\\nu} \\, T_{\\gamma}^{\\ \\gamma}\n \\right) n^{\\mu }\\xi^{\\nu} dV \n , \\label{komarM3}\n\\end{equation}\nwhere $M_H$ is the horizon mass of the black hole.\nThe mass of all (gravitating) solutions\ncan be directly obtained from the asymptotic form of their metric.\nIn the units employed, we find\n\\begin{equation}\nM= \\frac{1}{\\alpha^2}\\, \\lim_{r \\rightarrow \\infty} m(r)\n . \\label{mass} \\end{equation}\n\n\\section{Boson Stars}\n\n\\begin{figure}[p]\n\\begin{center}\n\\vspace{-1.5cm}\n\\mbox{\\hspace{-1.5cm}\n\\subfigure[][]{\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1a.eps}\n\\label{phasediag}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1b.eps}\n\\label{QB_r0_vs_b0}\n}\n}\n\\vspace{-0.5cm}\n\\mbox{\\hspace{-1.5cm}\n\\subfigure[][]{\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1c.eps}\n\\label{QB_M_vs_b0}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1d.eps}\n\\label{QB_Q_vs_b0}\n}\n}\n\\vspace{-0.5cm}\n\\mbox{\\hspace{-1.5cm}\n\\subfigure[][]{\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1e.eps}\n\\label{QB_om_vs_b0}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1f.eps}\n\\label{QB_Mvs_Q}\n}\n}\n\\end{center}\n\\caption{Properties of the boson star solutions shown versus $b(0)$,\nthe value of the gauge field function $b(r)$ at the origin:\n(a) $h(0)$, the value of the scalar field function $h(r)$ at the origin,\nforms with $b(0)$ the phase diagram of the solutions;\n(b) $r_o$ indicates the size of the solutions;\n(c) and (d) exhibit logarithmically\nthe mass $M$ and the charge $Q$, respectively;\nand (e) shows the value $b(\\infty)$ of the gauge field function\n$b(r)$ at infinity (which corresponds to the value\nof the scalar field frequency $\\omega$).\nIn (f) the ratio of mass and charge $M\/Q$ is exhibited versus \nthe charge $Q$.\nNote that $a=\\alpha^2$, and the asterisks mark the transition\npoints from boson stars ($Q$-balls) to $Q$-shells.\n\\label{bosonstar}\n}\n\\end{figure}\n\nWe first address the set of globally regular \nball-like boson star solutions,\nobtained by coupling the $Q$-ball solutions \\cite{Arodz:2008nm} to gravity.\nWe illustrate the physical properties of these solutions\nin Fig.~\\ref{bosonstar}.\n\nFig.~\\ref{phasediag} represents the phase diagram for the boson star\nsolutions. Here the sets of solutions for a sequence of values of the\ngravitational coupling $\\alpha$ are exhibited in terms of\nthe values of the matter field functions at the center of the solutions,\ni.e., the value of the scalar field function $h(0)$ \nand the value of the gauge field function $b(0)$.\nThe figure consists of four distinct regions,\nwhich we have labelled I, Ia, II and IIa.\nFor $\\alpha=0$, the $Q$-ball solutions form a continuous set,\nrepresented by a single curve in the lower part of the figure,\ndenoted as region I.\nThese non-gravitating solutions are\nbounded by some maximal value of $h(0)$, by some minimal value of $b(0)$,\nand by the bifurcation point with the shell-like solutions,\nwhere $h(0)$ reaches zero.\n\nAs the gravitational coupling constant $\\alpha$ is increased from zero, \nthe resulting sets of solutions smoothly deform and fill region I.\nThe maximal value of $h(0)$ \nof these sets of solutions (for each value of $\\alpha$ given by \na single continuous curve) then increases,\nthe corresponding minimal value of $b(0)$ decreases, \nand the bifurcation point with the shell-like solutions decreases as well.\nThis decrease of the bifurcation point continues until \nzero is reached at the critical value of \nthe gravitational coupling constant, $\\alpha_{sh} \\approx 0.1861$.\nBeyond $\\alpha_{sh}$ no shell-like solutions exist.\nHowever,\nthe set of solutions continues to deform smoothly with increasing $\\alpha$.\n\nThis smooth evolution continues, until a second \ncritical value of the gravitational coupling constant $\\alpha$ is reached,\n$\\alpha_{cr} \\approx 0.1989$.\nHere a bifurcation with a second set of solutions is encountered,\nwhich constitute the boundary of region Ia.\nThis second type of solutions is present for each finite value of\n$\\alpha \\le \\alpha_{cr}$. \nFor this type of solutions $h(0)$ has a minimal value for fixed $\\alpha$,\nwhich decreases with increasing $\\alpha$,\nuntil at $\\alpha_{cr}$ the bifurcation is reached.\n\nAt $\\alpha_{cr}$ the set I and set Ia\nsolutions touch and bifurcate. \nFor $\\alpha > \\alpha_{cr}$ \nthey then split into a right and left set of solutions,\nforming regions II and IIa, respectively.\nThe solutions in region II correspond to the larger values of $b(0)$,\nwhile the solutions in region IIa are restricted to the\nsmaller values of $b(0)$.\nWith increasing $\\alpha$, the sets of solutions in region IIa\nmove towards smaller values of $b(0)$,\npossibly disappearing at some critical value of the gravitational coupling,\nwhereas the sets of solutions in region II\nmove towards larger values of $b(0)$.\n\nFig.~\\ref{QB_r0_vs_b0} shows the outer radius $r_o$ for these sets of solutions,\nand thus the size of the corresponding boson stars.\nClearly, the biggest size for a given $\\alpha \\le \\alpha_{cr}$\nis always reached in region I at the bifurcation point\nwith the shell-like solutions.\nThe oscillations of the gauge field value $b(0)$ \nwith increasing scalar field value $h(0)$ seen in region II in\nFig.~\\ref{phasediag}\nare reflected in the spirals\nformed by the outer radius $r_o$ in region II in Fig.~\\ref{QB_r0_vs_b0}.\nThey are also present in regions IIa and Ia, whenever\nthe gauge field value $b(0)$ exhibits oscillations.\n\nThe mass $M$ and the charge $Q$ of these sets of boson star solutions\nare exhibited in Figs.~\\ref{QB_M_vs_b0} and \\ref{QB_Q_vs_b0}.\nBoth show a very similar pattern.\nAgain, the biggest mass and charge for a given $\\alpha \\le \\alpha_{cr}$\nare reached in region I at the respective bifurcation point\nwith the shell-like solutions,\nwhile the oscillations of $b(0)$ seen in regions Ia, II and IIa\nlead to spiral patterns for the mass and charge.\n\nThe corresponding family of curves for the asymptotic\nvalue $b(\\infty)$ of the gauge field function $b(r)$ at infinity\n(which can be indentified with the value of the\nscalar field frequency $\\omega$ in the gauge,\nwhere the gauge field vanishes at infinity)\nis exhibited in Fig.~\\ref{QB_om_vs_b0}.\nHere the overall pattern is different, but spirals occur as well.\nFinally, in Fig.~\\ref{QB_Mvs_Q} we exhibit\nthe ratio of mass and charge $M\/Q$ versus $Q$.\nWe observe a linear increase of $M\/Q$ with $Q$ for the larger values\nof $Q$ in regions I and IIa, where\nthe slope decreases with increasing $\\alpha$,\nmaking $M\/Q$ almost constant for larger values of \n$\\alpha$ (e.g.,~$\\alpha=0.22$).\n\nWhile the occurrence of spirals is a typical feature of\nboson star solutions \\cite{Friedberg:1976me}, \nthe present sets of solutions exhibit a for boson stars new phenomenon,\nnamely the formation of throats.\nAs a throat is formed,\nthe minimum of the metric function $N(r)$ tends to zero,\nand the zero is reached precisely at the outer radius $r_o$.\nAt the same time the metric function $A(r)$ tends to a step\nfunction, that vanishes inside $r_o$, and assumes the asymptotic value\n$A(r)=1$ outside $r_o$.\n(In Fig.~\\ref{fun} the functions close to throat formation\nare exhibited in the case of black holes.)\n\nThe space-time for $r \\ge r_o$ then corresponds to the exterior\nspace-time of an extremal Reissner-Nordstr\\\"om (RN) black hole.\nIndeed, there the metric function $N(r)$ can be expressed as\n\\begin{equation}\nN(r) = 1 - \\frac{2 \\alpha^2 M}{r} + \\frac{\\alpha^2 Q^2}{r^2} \n = \\left( 1 - \\frac{\\alpha Q}{r} \\right)^2 \n , \\label{RN} \\end{equation}\ni.e., $r_H = r_o= \\alpha^2 M = \\alpha Q$ for the extremal RN solution\n(in the units employed).\nAs seen in Fig.~\\ref{QB_Mvs_Q},\nthis relation is precisely satisfied, when $b(0) \\rightarrow 0$.\nThus a throat is formed, when in a set of solutions the value $b(0)$ \nof the gauge field function tends to zero.\nIn fact, the function $b(r)$ then tends to zero in the whole region\n$r< r_o$, and its derivative $b'(r)$ does so as well.\nHowever, at $r_o$ the derivative $b'(r)$ jumps to a finite\nvalue, necessary for the \nCoulomb fall-off of a solution with charge $Q$.\n\nFinally we note, that the sets of boson star solutions\nwith fixed gravitational coupling constant $\\alpha$ \nsatisfy a mass relation.\nThis relation is based on the observation, that\n\\begin{equation}\nd M = b(\\infty) d Q \n , \\label{Mreg2} \\end{equation}\nshown to hold for the regular solutions in flat space \\cite{Arodz:2008nm}.\nSince (\\ref{Mreg2}) continues to hold for the gravitating solutions,\nintegration \nyields the mass relation\n\\begin{equation}\nM_2 =  M_1 + M_Q =\n M_1 + \\int_{Q_1}^{Q_2} b(\\infty) d Q\n , \\label{Mreg} \\end{equation}\nwhere the mass $M_2$ of a regular solution with charge $Q_2$\nis obtained \nby integrating from any regular solution $M_1$ with charge $Q_1$\nalong the curve of intermediate solutions of the set.\n\n\n\n\\section{Gravitating $Q$-Shells}\n \n\\begin{figure}[t!]\n\\begin{center}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig2a.eps}\n\\label{fun1}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig2b.eps}\n\\label{fun2}\n}\n}\n\\end{center}\n\\caption{Functions of the gravitating $Q$-shell solutions \nshown versus the radial coordinate $r$ for $Q=10$ and $\\alpha^2=0.1$:\n(a) metric functions $A(r)$ and $N(r)$;\n(b) matter functions $h(r)$ and $b(r)$.\nAlso shown are the corresponding functions for\nblack hole solutions with several horizon radii $r_H$.\nThe largest $r_H$ is close to the critical\nvalue, where the throat is formed.\n\\label{fun}\n}\n\\end{figure}\n\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig3a.eps}\n\\label{phaseSdiag2}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig3b.eps}\n\\label{QS_r0_vs_b1}\n}\n}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig3c.eps}\n\\label{QBS_M_vs_b1}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig3d.eps}\n\\label{QS_MQ_vs_b1}\n}\n}\n\\end{center}\n\\caption{Properties of the gravitating $Q$-shell solutions shown versus \n$b(r_i)$,\nthe value of the gauge field function $b(r)$ at the inner \nshell radius $r_i$:\n(a) $r_i\/r_o$, the ratio of inner and outer shell radii;\n(b) the outer shell radius $r_o$;\n(c) the mass $M$ of shell-like solutions and boson stars \n(resp.~$Q$-balls for $\\alpha=0$);\n(d) the scaled mass $\\alpha^2 M$ and the scaled charge $\\alpha Q$.\nNote that $a=\\alpha^2$, and the asterisks mark the transition\npoints from boson stars ($Q$-balls) to $Q$-shells.\n\\label{Q_shell}\n}\n\\end{figure}\n\nLet us next consider the gravitating shell-like solutions.\nHere the space-time consists of 3 parts.\nIn the inner part $0  \\le r < r_i$ \nthe gauge potential is constant and the scalar field vanishes.\nConsequently, it is Minkowski-like,\nwith $N(r)=1$ and $A(r)={\\rm const}<1$. \nThe middle region $r_i < r < r_o$ then represents the shell of\ncharged bosonic matter, while the outer region\n$r_o < r < \\infty$ corresponds to part of a Reissner-Nordstr\\\"om\nspace-time, where the gauge field exhibits the standard Coulomb fall-off,\nwhile the scalar field vanishes identically.\nThis behaviour of the functions is demonstrated in Fig.~\\ref{fun} for\nthe shell-like solution with charge $Q=10$ and \ngravitational coupling constant $\\alpha^2=0.1$.\n\nWe exhibit in Fig.~\\ref{Q_shell} the properties of\nshell-like solutions.\nFig.~\\ref{phaseSdiag2} shows the ratio of the\ninner radius $r_i$ to the outer radius $r_o$ for these solutions.\nFor a given finite value of the gravitational coupling,\nthe branch of gravitating shells emerges\nat the corresponding boson star solution and ends,\nwhen a throat is formed at the outer radius $r_o$.\nAs this happens, the value of\n$b(r_i)$ reaches zero (or equivalently $b(0)\\rightarrow 0$,\nsince $b(r)$ is constant in the interior, $0 \\le r \\le r_i$).\nThe exterior space-time $r > r_o$ then corresponds to the exterior of an\nextremal RN space-time.\n\nThus in contrast to $Q$-shells in flat space, which grow rapidly\nin size, mass and charge as the ratio $r_i\/r_o \\rightarrow 1$,\nthe growth of gravitating $Q$-shells is limited by gravity,\nand the restriction in size, mass and charge is the stronger, \nthe greater the value of the gravitational coupling constant $\\alpha$.\nThis is demonstrated in Figs.~\\ref{QS_r0_vs_b1}, \\ref{QBS_M_vs_b1}\nand \\ref{QS_MQ_vs_b1}, \nwhere the outer radius $r_o$, the mass $M$\nand the charge $Q$ are exhibited\nfor a sequence of values of the gravitational coupling constant.\n\nIn Fig.~\\ref{QBS_M_vs_b1} for comparison also the mass of the\ncorresponding boson star solutions (resp.~$Q$-ball solutions\nfor vanishing gravitational coupling constant) are exhibited.\nThe transitions from the ball-like to the shell-like solutions\nare indicated in the figure by the small asterisks.\nWith increasing $\\alpha$ the sets of shell-like solutions\ndecrease rapidly, until at the critical value\n$\\alpha_{sh}$ (see Fig.~\\ref{phasediag})\nthey cease to exist.\n\nFig.~\\ref{QS_MQ_vs_b1} exhibits the scaled mass $\\alpha^2 M$\nand the scaled charge $\\alpha Q$ for several sets of\ngravitating $Q$-shells. Together with Fig.~\\ref{QS_r0_vs_b1}\nthe figure demonstrates,\nthat the condition \nfor extremal RN solutions,\n$r_o= \\alpha^2 M = \\alpha Q$,\nis satisfied\nfor the shell-like solutions,\nas the throat forms at the outer radius $r_o$.\n\nFinally we note, that the shell-like solutions satisfy the mass relation\n(\\ref{Mreg}) as well.\nConsequently the mass relation holds for any two globally regular solutions\nof a set with given gravitational coupling constant,\nthus relating also ball-like and shell-like solutions.\n\n\\section{Black Holes}\n\n\\begin{figure}[p!]\n\\begin{center}\n\\vspace{-1.0cm}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4a.eps}\n\\label{BHQ10_b1_vs_rh}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4b.eps}\n\\label{BHQ100_b1_vs_rh}\n}\n}\n\\vspace{-0.5cm}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4c.eps}\n\\label{BHQ10_M_vs_rh}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4d.eps}\n\\label{BHQ100_M_vs_rh}\n}\n}\n\\vspace{-0.5cm}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4e.eps}\n\\label{BHQ10_T_vs_rh}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4f.eps}\n\\label{BHQ100_T_vs_rh}\n}\n}\n\\end{center}\n\\caption{Properties of the black hole solutions \nwith Schwarzschild-type interior shown versus \nthe horizon radius $r_H$:\nThe left column exhibits for solutions with fixed charge $Q=10$\n(a) \n$b(r_i)$, the value of the gauge\nfield function $b(r)$ at the inner shell radius $r_i$;\n(c) the mass $M$;\n(e) the ratio of the temperature $T$ at the black hole horizon $r_H$ to the\ncorresponding Schwarzschild black hole temperature $T_S$.\nThe right column ((b), (d), (f)) exhibits the\nsame for solutions with fixed charge $Q=100$.\nNote that $a=\\alpha^2$. \n\\label{BH1}\n}\n\\end{figure}\n\nLet us finally address black holes in this model.\nThe simplest type of black holes is obtained,\nwhen the Minkowski-like inner part of the space-time,\n$0 \\le r \\le r_i$, of gravitating $Q$-shell solutions\nis replaced by the inner part of a curved Schwarzschild-like space-time.\nThe metric in the interior region $0 \\le r \\le r_i$\nis then determined by the function $N(r)= 1 - (r_H\/r)$ \nand a constant function $A(r)$.\nThus the event horizon resides at $r_H < r_i$.\nBut the presence of the $Q$-shell outside the event horizon,\nmakes the properties of the black hole differ from those\nof a pure Schwarzschild black hole.\n\nSince with the event horizon size a further variable appears,\nwhich is an important physical quantity,\nwe discuss the black hole properties with respect to the\nhorizon radius $r_H$ in the following.\nThe metric and matter field functions \nfor black holes with charge $Q=10$ \nat gravitational coupling $\\alpha^2=0.1$\nare exhibited in Fig.~\\ref{fun} \nfor several values of the horizon radius $r_H$.\n\nTo illustrate the domain of existence of such black hole solutions,\nwe again choose a sequence\nof values for the gravitational coupling constant,\nbut we now keep the charge $Q$ fixed, as we vary \nthe horizon radius, starting from the corresponding\nglobally regular $Q$-shell solution.\nA respective set of solutions is shown in Fig.~\\ref{BH1}\nfor $Q=$10 and 100.\n\nFirst of all we note, that the horizon radius is always limited\nin size, where the maximal size grows with the charge $Q$.\nFor small $Q$, e.g.~$Q=$10, we observe two distinct patterns\nfor the black hole solutions.\nThe first pattern arises when the\nfixed gravitational coupling constant has a value\nbelow a certain critical value.\nHere a maximal horizon size is reached,\nwhen the horizon radius $r_H$ gets close to the inner radius \nof the shell $r_i$.\nThere a bifurcation occurs and a second branch emerges,\nwhich ends at a second bifurcation, where a third branch emerges, etc.\nThis results in a spiralling pattern,\nwhere the mass $M$ and the temperature $T$ of the solutions tend towards\nfinite limiting values.\n(The first few branches are apparent in Fig.~\\ref{BHQ10_b1_vs_rh},\nand enlarged in the inlet for a representative value\nof the gravitational coupling constant, $\\alpha^2=0.01$,\nwhile the higher branches are too small to be resolved there.)\n\nThe second pattern is present above that critical value\nof the coupling constant. Here the set of black hole solutions\nfor fixed gravitational coupling ends, when a throat is formed\nat the outer shell radius $r_o$.\nThere the condition for extremal RN solutions,\n$r_o= \\alpha^2 M = \\alpha Q$, is satisfied again,\nas seen in Figs.~\\ref{BHQ10_M_vs_rh} and \\ref{BHQ100_M_vs_rh}.\nAs the throat forms,\nthe temperature $T$ at the event horizon of the Schwarzschild-like\nblack hole $r_H < r_i$ tends to zero,\nas seen in Figs.~\\ref{BHQ10_T_vs_rh} and \\ref{BHQ100_T_vs_rh}.\n\nWhile appearing at first unexpected,\nthe reason for the vanishing of the temperature $T$\nis the behaviour of the metric function $A(r)$\nin $g_{tt}$, since $A(r)$ tends to zero in the interior,\nwhen the throat is formed,\nas seen in Fig.~\\ref{fun1}.\nWe recall, that the ratio of the temperature $T$ of\nthe black hole within the $Q$-shell \nto the temperature \n$T_{\\rm S} = (4 \\pi r_{\\rm H})^{-1}$ \nof the Schwarzschild black hole is given by\n$\\displaystyle T \/ T_{\\rm S}\n  =  \\left. r A N' \\right|_{r_{\\rm H}} = A(r_H)$.\n\nFor larger (fixed) values of the charge we always observe\nthis second pattern,\nalthough the throat may either be reached directly after a monotonic\nincrease of the horizon radius $r_H$ to its maximum value,\nor along a second branch, where the\nhorizon radius is decreasing again (having passed a bifurcation),\nas seen in Fig.~\\ref{BHQ100_b1_vs_rh}.\n\nAs seen in the figure,\nwhenever bifurcations occur, there are two (or more) black hole\nspace-times with the same value of the charge $Q$ and the\nsame horizon radius $r_H$ (within a certain range of values),\nbut different values of the total mass $M$\nas measured at infinity.\nSurprisingly, however, \nthere are also two (or more) black hole space-times\nwith the same value of the charge $Q$ and the same \nvalue of the total mass $M$ (within a certain range of values).\nThese black holes thus have the same set of global charges\nbut are otherwise distinct solutions of the Einstein-matter equations.\nConsequently black hole uniqueness does not hold in this model\nof scalar electrodynamics.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\vspace{-0.5cm}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig5a.eps}\n\\label{figiso1}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig5b.eps}\n\\label{figiso2}\n}\n}\n\\end{center}\n\\caption{Mass formulae for black hole solutions:\n(a) the mass $M$ of solutions with fixed charge $Q=10$ and fixed\ngravitational coupling constant $\\alpha$,\nobtained from the asymptotic metric (\\ref{mass}) and the\nmass formula (\\ref{IHmu});\n(b) the mass $M$ of solutions with variable charge $Q$ but\nfixed ratio of inner and outer shell radii $r_i\/r_o$ for several fixed\nvalues of the gravitational coupling constant $\\alpha$,\nobtained from the asymptotic metric (\\ref{mass}) and the\nmass formula (\\ref{IHmuDQ2}).\nNote that $a=\\alpha^2$.\n\\label{BH2}\n}\n\\end{figure}\n\nLet us finally consider some mass relations\nfor these black holes space-times possessing $Q$-shells.\nWe begin by recalling an interesting result\nobtained in the isolated horizon framework\n\\cite{Ashtekar:2004cn}.\nIt states that the mass $M$ \nof a black hole space-time with horizon radius $r_H$\nand the mass $M_{\\rm reg}$\nof the corresponding globally regular space-time\nobtained in the limit $r_H \\rightarrow 0$ are related via\n\\cite{Corichi:1999nw,Ashtekar:2000nx,Ashtekar:2004cn}\n\\begin{equation}\nM = M_{\\rm reg} + M_\\Delta  ,\n\\label{IHmu} \\end{equation}\nwhere the mass contribution $M_\\Delta$ is defined by\n\\begin{equation}\nM_\\Delta = \\frac{1}{\\alpha^2} \\, \\int_0^{r_H} \\kappa(r'_H)r'_H d r'_H  .\n\\label{IHmuD}\n\\end{equation}\nHere $\\kappa(r_H)$ represents the surface gravity \nof the black hole with horizon radius $r_H$,\n$\\kappa = 2 \\pi T$.\nAccordingly, the mass $M$ \nof a space-time with a black hole\nwith horizon radius $r_H$ within a $Q$-shell with total charge $Q$\nshould be obtained as the sum of the globally regular gravitating $Q$-shell\nwith charge $Q$ and the integral $M_\\Delta$ along the set of black hole\nspace-times, obtained by increasing the horizon radius for fixed charge\nfrom zero to $r_H$.\n\nThis relation is demonstrated in Fig.~\\ref{figiso1}\nfor the set of solutions with charge $Q=10$ and gravitational coupling\nconstant $\\alpha^2=0.01$.\nThe values for the mass $M$\nobtained from the relation (\\ref{IHmuD}) are seen to agree with\nthe values for the black hole mass $M$ obtained from the\nasymptotics (\\ref{mass}).\nThe set of solutions exhibited has spiralling character,\ni.e., it has besides the main first branch\na second branch, also exhibited, and further branches, \nnot resolved in the figure.\n\nWhen the charge is allowed to vary, too, one expects\na change of the above relation in accordance with (\\ref{Mreg2})\nand the first law (in the units empoyed), i.e.,\n\\begin{equation}\ndM = \\frac{\\kappa}{8 \\pi \\alpha^2} d{\\cal A} + b(\\infty) dQ\n , \\label{firstlaw} \\end{equation}\nwhere ${\\cal A} = 4 \\pi r_H^2$ denotes the area of the horizon\nand $b(\\infty)$ represents the electrostatic potential at infinity.\nThus we generalize the above relation (\\ref{IHmu}) to read\n\\begin{equation}\nM = M_{\\rm reg} + M_\\Delta + M_Q =\n M_{\\rm reg} + M_\\Delta \n  +  \\int_{Q_{\\rm reg}}^{Q} b(\\infty) d Q' .\n\\label{IHmuDQ2}\n\\end{equation}\nThis relation is demonstrated in Fig.~\\ref{figiso2},\nwhere for several values of the gravitational coupling constant\nand for fixed ratio\nof inner and outer shell radii $r_i\/r_o$,\nthe values for the mass $M$\nobtained from the relation (\\ref{IHmuDQ2}) are shown together with\nthe values for the mass $M$ obtained from the\nasymptotics (\\ref{mass}).\n\n\n\\section{Conclusion and Outlook}\n\nWe have considered boson stars, gravitating $Q$-shells and\nblack holes within $Q$-shells in scalar electrodynamics\nwith a $V$-shaped scalar potential,\nwhere the scalar field is finite only in compact ball-like or\nshell-like regions.\n\nThe gravitating $Q$-shells surround a flat Minkoswki-like interior region,\nwhile their exterior represents part of an exterior\nRN space-time.\nWhen the flat interior is replaced by a Schwarzschild-like\ninterior, black hole space-times result, where\na Schwarzschild-like black hole is surrounded by a compact shell of\ncharged matter, whose exterior again represents part of an exterior\nRN space-time.\n\nThese black hole space-times violate black hole uniqueness,\nin certain regions of parameter space.\nHere for the same values of the mass $M$ and the charge $Q$\ntwo or more distinct solutions of the Einstein-matter equations\nexist.\n\nThe solutions satisfy certain relations of the type obtained first in the\nisolated horizon formalism, which connect the mass $M$ of \na black hole solution with the mass $M_{\\rm reg}$ of the\nassociated globally regular solution.\nThe masses of two regular solutions are related in an analogous \n(simpler) manner.\nThis formalism further suggests to interpret the\nblack hole space-times as bound states of\nSchwarzschild-type black holes and gravitating $Q$-shells\n\\cite{Ashtekar:2000nx}.\n\nWhile we have restricted our discussion here to\nSchwarzschild-type black holes in the interior,\nthere are also black hole space-times with charged, i.e.,\nReissner-Nordstr\\\"om-type interior solutions.\nThese more general black hole space-times \nwill be discussed elsewhere.\n\nThe inclusion of rotation presents another interesting generalization\nof the solution considered here, \nsince rotating boson stars are well-known \n\\cite{Mielke:2000mh,Schunck:2003kk,Yoshida:1997qf,Kleihaus:2005me,Kleihaus:2007vk}.\nThe construction of the corresponding rotating shells\nand their black hole generalizations, however, still poses a challenge.\n\n\\vspace{0.5cm}\n{\\bf Acknowledgement}\n\n\\noindent\nBK gratefully acknowledges support by the DFG,\nCL and ML by the DLR.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHurwitz theory is the study of maps of algebraic curves, viewed as ramified covers of orientable surfaces. At the intersection of geometry, representation theory and combinatorics, it is an area that naturally lends itself to making bridges and connections. In this paper we study the combinatorial structure of certain Hurwitz spaces via a parallel investigation in tropical geometry. \n\n\n\\subsection{Summary}\n\nThe study of the connections between classical and tropical Hurwitz theory was initiated in \\cite{CJM10} and \\cite{CJMwc};  the tropical point of view provided a combinatorial interpretation of double Hurwitz numbers which was very well tuned to describing the polynomial aspects and the wall crossing phenomena occurring in the theory. We  continue this exploration by studying rational double Hurwitz loci inside spaces of (relative\/tropical) stable maps, generically parameterizing covers of $\\mathbb{P}^1$ with two prescribed ramification profiles and a part of the branch divisor fixed, and their pushforwards to the moduli space of curves, which we call double Hurwitz cycles. Besides the genus, the discrete invariants we fix are the total length $n$ of the special ramification profile, and the dimension of the loci we want to study. Then we study families of Hurwitz loci parameterized by integral points in an $(n-1)$ dimensional vector space. \n\nOn the classical side,  we realize Hurwitz loci as the pullback via a natural branch morphism of appropriate strata in spaces of pointed  chains of projective lines (Losev-Manin spaces, Section \\ref{sec:mscm}). This gives a boundary expression for Hurwitz cycles where the coefficients are piecewise polynomials in the entries of the special ramification data (Theorem \\ref{thm:poly}).  \n\nOn the tropical side, we realize the Hurwitz loci as tropical Gromov-Witten cycles. Our tropical Hurwitz cycles are balanced polyhedral complexes; their topology is constant in the chambers of polynomiality of (classical) Hurwitz cycles, whereas their geometry (affine integral structure, weights and coordinates of vertices) varies in a polynomial way in terms of the special ramification profiles. \n\nNaturally, we also study the connection between classical and tropical Hurwitz cycles (Section \\ref{sec:tcc}) and observe a natural combinatorial duality between tropical and classical strata. To be more precise, the stratification on the tropical side is the polyhedral structure inherited from the moduli space of tropical curves. The stratification on the classical side is the usual stratification in boundary classes. For Hurwitz cycles of dimension $d$, $k$-dimensional classical strata correspond to $d-k$-dimensional tropical strata, and the combinatorial type of the tropical stratum can be encoded in terms of the dual graph of the classical stratum.\n\n\nWe conclude the paper by studying how Hurwitz cycles vary across the walls of the chambers of polynomiality, both on the classical and on the tropical side. In a similar fashion to Hurwitz numbers, the wall crossing  formulae have an inductive  form:  the cycles in the formula can be obtained as pushforwards via appropriate gluing morphism  of pairs of boundary strata  coming from Hurwitz cycles where the profile data is split according to the equation of the wall, and the dimensions are split in all possible ways adding up to the correct one. \nEven though the final form of the tropical and classical wall-crossing formulae is essentially the same, there are some subtleties involved in even making sense of what a tropical wall crossing formula may be: that's why we treat the two cases separately. The classical wall crossing formula is Theorem \\ref{thm:wc}, the (cleanest form of  the) tropical one Corollary  \\ref{cor:twc}.\n\nTo make our exposition easier to follow, throughout the paper we use the one-dimensional case ((tropical) Hurwitz curves) as a running example.\n\n\\subsection{Context, Motivation and further directions of our research}\n\nBecause of the many and diverse categories equivalent to curves and their maps,  Hurwitz theory is by nature ``interdisciplinary'': exploring the dictionary between the tropical and classical approaches to Hurwitz theory is a natural thing to do. It has already been fruitful and hopefully will bear even more applications in the future. \n\nBefore Hurwitz theory made its appearance in tropical geometry, the area of tropical enumerative geometry was pioneered by Mikhalkin's celebrated Correspondence Theorem which relates classical numbers of plane curves to their tropical counterparts \\cite{Mi03}. Nowadays, numbers of tropical curves can (at least in the rational case) be understood as intersection products in an appropriate moduli space of tropical curves, just as in the classical world. For higher genus, understanding the appropriate moduli space of tropical curves and its connection to the moduli space of algebraic curves is an active area of research (see e.g.\\ \\cites{CMV12,Cap12a}). Also in the rational case, the connection between the intersection theory of the moduli space of algebraic curves and the moduli space of tropical curves is not yet completely understood. Combinatorial dualities such as the one we observe in Section \\ref{sec:tcc} are present in many situations, but in our opinion they do not fully explain the success of tropical methods in enumerative geometry. We expect deeper connections to be discovered. Our paper contributes some interesting and geometrically meaningful families of cycles in the intersection ring of tropical $\\mathcal{M}_{0,n}$, and makes important steps in understanding the correspondence of these cycles to their classical counterparts. We hope to extent the study to higher genus, and to contribute to the understanding of the deeper connections between moduli spaces of algebraic and tropical curves.\n\n\nClassically double Hurwitz loci were a key ingredient in the proof of  Theorem $\\star$, the main result of  \\cite{gv:rvl}: tautological classes in the moduli space of curves of degree greater than $g-1$ (say $g+k$ with $k$ a non-negative integer) must admit an expression supported on the boundary, and more specifically on strata parameterizing curves with at least $k$ rational components. Then again they were applied to the study of tautological classes in \\cite{gjv:last}, even though in neither of these cases they were viewed as families of loci with any sort of polynomial structure. This makes its appearence for the case of $0$-dimensional cycles, or more mundanely double Hurwitz numbers, in \\cite{gjv:ttgodhn}. After \\cites{ssv:cbodhn,CJM10,CJMwc} the algebro-combinatorial aspects of the piecewise polynomiality\nof double Hurwitz numbers are well understood, leading the way to some really interesting geometric questions: \n\\begin{description}\n\\item[ELSV-type formula for double Hurwitz numbers] can double Hurwitz numbers be obtained as intersections of tautological classes on some family of birational moduli spaces --- constant in the polynomiality chambers ---  in a way that naturally explains the structure of the polynomials? Can the wall crossings be somehow seen as coming from the birational transformations occurring when crossing the walls?\n\\item[Higher dimensional Hurwitz loci] How well does the piecewise polynomial structure carry over to higher dimensional loci? In particular can we understand full Hurwitz spaces (or rather their compactifications such as Admissible Covers or Relative Stable Maps) as tautological classes in the moduli space of curves?\n\\end{description}\nThis paper provides an exhaustive answer for the second question in genus $0$: here the full moduli space is birational to $\\overline{M}_{0,n}$, and we show that every time we  increase the codimension by one by fixing a simple branch point in the branch divisor we obtain a polynomial class of one degree higher. In genus $0$ an ELSV formula  is trivially showed to hold in the one part double Hurwitz number case (\\cite{gjv:ttgodhn}), and we are hopeful that a geometric understanding of the wall crossings will be complete soon. In higher genus, the situation is both more complicated and more interesting: here the Hurwitz moduli space represents a codimension $g$ tautological class, which has recently been at the center of attention because of its connections with symplectic field theory (\\cite{eliashberg}).  Recently Hain (\\cite{H11}) has produced  tautological  classes on $\\overline{M}_{g,n}$,  which agree with double Hurwitz loci when restricted to the partial compactification of curves of compact type (however simple intersection computations show that Hain's class does not agree with either the Admissible Cover nor the Relative Stable Map compactification of the Hurwitz space already in genus one). Interestingly, Hain's class is homogeneous polynomial of degree $2g$. Understanding how Hain's class compares with Admissible Covers or Relative Stable Maps, besides being a very interesting question on its own, is likely a useful ingredient in the quest for an ELSV formula for double Hurwitz numbers. Our work is aiming in that direction in a couple different ways. On the one hand we interpolate a family of classes  starting from the zero dimensional loci which we understand very well. On the other hand we make a connection with tropical geometry, which is usually well tuned to give information about the  deeper boundary strata of the classical moduli spaces of curves.\n\n\n\n\\subsection{Acknowledgements}\nThis work is the result of a two week long \\textit{Research in Pairs} at the Oberwolfach Institute for Mathematics, which the authors thank for its hospitality. The second author was at the intersection of supports by a Simons Collaboration Grant and NSF grant  DMS110549. The third author was partially supported by DFG-grant MA 4797\/1-2.\n \n\n\n\\section{Background and Notation}\nIn this section we recall the basic definitions and constructions that are needed for the set-up of the theory. \n\\subsection{Moduli Spaces of Curves and  Maps}\n\\label{sec:mscm}\n\nWe assume that the reader is familiar with $\\overline{M}_{0,n}$, the moduli space of rational pointed stable curves, a smooth projective variety of dimension $n-3$. The Chow ring of $\\overline{M}_{0,n}$ is generated by irreducible boundary divisors, with the only relations (besides the obvious ones given by empty intersections) generated by the WDVV relations (\\cite{sk:m0n}). Irreducible boundary strata are identified by their dual graph: given a graph $\\Gamma$, we denote the corresponding stratum by $\\Delta_\\Gamma$.\n\nIn \\textit{weighted stable curves} one tweaks the stability of a rational pointed curve $(X= \\cup_j X_j, p_1, \\ldots, p_n)$ by assigning weights $a_i$ to the marked points and requiring the restriction to each $X_j$ of  $\\omega_X +\\sum a_ip_i$ to be ample (this amounts to the combinatorial condition that  $\\sum_{p_i\\in X_j} a_i + n_j>2$, where $n_j$ is the number of shadows of nodes on the $j$-th component of the normalization of $X$). \n\nWhen two points are given weight $1$ and all other points  very small weight, the space $\\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})$ is classically known as the \\textit{Losev-Manin} space {\\cite{lm:lms}}: it parameterizes chains of $\\mathbb{P}^1$'s with the heavy points on the two terminal components and light points (possibly overlapping amongst themselves) in the smooth locus of the chain.\n\nLet  $\\mathbf{x}$ be an $n$-tuple  of integers adding to $0$, and denote $\\mathbf{x^+}$ (resp. $\\mathbf{x^-}$) the sub-tuple of positive (resp. negative) parts. We consider moduli space of relative stable maps $\\overline{M}_0(\\mathbb{P}^1; \\mathbf{x^+}0, \\mathbf{x^-}\\infty)$ and their ``rubber'' variant  $\\overline{M}^\\sim_0(\\mathbb{P}^1; \\mathbf{x^+}0, \\mathbf{x^-}\\infty)$ (see \\cites{gv:rvl,mp:tvgwt}).  An important technical detail is we mark the preimages of the relative points. In order to mark some of the simple ramification points on the source curve, we introduce a space of relative weighted stable maps, where in addition to $0$ and $\\infty$ there are $j$ moving simple transposition points that are marked and given weight $\\varepsilon$.  These spaces  are typically denoted $\\overline{M}^\\sim_0(\\mathbb{P}^1; \\mathbf{x^+}0, \\mathbf{x^-}\\infty, \\varepsilon t_1, \\ldots, \\varepsilon t_j)$. \n\\begin{note}\nIn all our spaces of maps we make the notation lighter by  forgetting the target (always $\\mathbb{P}^1$),  noting that $\\mathbf{x}$ gives sufficient information to   determine the  relative points fixed at $0 $ and $\\infty$ and that the additional transposition points are understood to be ``light''. For example we write $\\overline{M}^\\sim_0(\\mathbf{x}, t_1, \\ldots, t_j)$ for $\\overline{M}^\\sim_0(\\mathbb{P}^1; \\mathbf{x^+}0, \\mathbf{x^-}\\infty, \\varepsilon t_1, \\ldots, \\varepsilon t_j)$. \n\\end{note}\n\n\nThere is a natural stabilization map $\\st$ to $\\overline{M}_{0,n}$ that forgets the map and remembers the (marked) points over $0$ and $\\infty$, and a branch map to an appropriate quotient of a Losev-Manin space, recording the position of the $r+2=n$ branch points. \nSince a degree $0$ divisor on $\\mathbb{P}^1$ determines a rational function up to a multiplicative constant, the map $\\st: \\overline{M}^\\sim_0(\\mathbf{x}) \\to \\overline{M}_{0,n}$ is birational. Marking $j$ simple transpositions makes $\\st$ into a degree ${{r}\\choose{j}}$ cover. The branch map is a cover of the Losev-Manin space of degree the double Hurwitz number $H_0(\\mathbf{x})$. \n\n\\subsubsection{Multiplicities of boundary strata.}\nBoundary strata in moduli spaces of relative stable maps corresponding to breaking the target are naturally described in terms of products of other moduli spaces of relative stable maps. It is important to keep careful track of various multiplicities coming both from combinatorics of the gluing and infinitesimal automorphisms (see \\cite{gv:rvl}*{Theorem 4.5}). \nLet $S$ be a boundary stratum in $\\overline{M}^\\sim_0(\\mathbf{x})$, parameterizing maps to a chain $T^N$  of $N$ projective lines. $S$ can be seen as the image of:\n$$\n\\gl: \\prod_{i=1}^N \\mathcal{M}^\\bullet_i \\to S \\subset \\overline{M}^\\sim_0(\\mathbf{x}),\n$$\n\nwhere the $\\mathcal{M}^\\bullet_i$ are moduli spaces of possibly disconnected relative stable maps, where the relative condition imposed at the point $\\infty$ in the $i$-th line matches the condition at $0$ in the $(i+1)$-th line. We denote by $\\mathbf{z_i}=(z_i^1, \\ldots, z_i^{r_i})$ such relative condition and by abuse of notation we say it is the relative condition at the $i$-th node of $T^N$. Then,\n\\begin{equation}\n\\label{bound:multi}\n[S] = \\prod_{i=1}^{N-1}\\frac{\\prod_{j=1}^{r_i} z_i^j}{|\\Aut(\\mathbf{z_i})|} \\left[\\gl_\\ast\\left( \\prod_{i=1}^N \\mathcal{M}^\\bullet_i \\right)\\right].\n\\end{equation}\n\nEquation \\eqref{bound:multi} seems horrendous, but it amounts to the following recipe: the general element in $S$ is represented by a map from a nodal curve  $X$ to $T^N$, with matching ramification on each side of each node of $X$. The multiplicity $m(S)$ is the product of ramification orders for each node of $X$ divided by the (product of the order of the group of) automorphisms of each partition of the degree prescribing the ramification profile over each node of $T^N$.\n\n\n\n\n\n\\subsection{Tropical Geometry}\n\\label{sec:tg}\nWe assume that the reader is familiar with tropical varieties and tropical cycles in a vector space with a lattice, i.e.\\ with weighted polyhedral complexes (possibly with negative weights in the case of cycles) satisfying the balancing condition around each cell of codimension one.\nThe exact polyhedral complex structure is not important. We do not distinguish between equivalent tropical cycles, i.e.\\ cycles which allow a common refinement respecting the weights. \n\n A rational function\n  on a tropical cycle is a continuous function that is\n  affine on each cell, and whose linear part is\n  integer. To a tropical cycle $X$ and a rational function $\\varphi$, we can associate the divisor $\\varphi\\cdot X$, a tropical subcycle of codimension one supported on\n  the subset of $X$ where $\\varphi$ is not linear \\cite{AR07}*{Construction 3.3}.\nWe can also form multiple intersection products $\\varphi_1 \\cdot \\ldots \\cdot\n  \\varphi_m \\cdot X$. They are commutative by \\cite{AR07}*{Proposition 3.7}. The weights of cells of  intersection products can be computed locally as follows.\n  \n\\begin{rem}\\label{rem-intersect}\n  Let $h_1, \\ldots, h_m$ be linearly independent integer linear\n  functions on $\\mathbb{R}^n$. By $H : \\mathbb{Z}^n \\rightarrow \\mathbb{Z}^m $ we\n  denote the linear map given by $x \\mapsto (h_1(x), \\ldots, h_m(x))$. Consider\n  the rational functions $ \\varphi_i = \\max\\{h_i,p_i\\}$ on $\\mathbb{R}^n$, where $p_i$ are fixed constants in $\\mathbb{R}$. These rational functions give rise\n  to an intersection product, which obviously consists of only one cone with \n weight equal to the order of the torsion part of $ \\mathbb{Z}^m \/ \\text{Im}(\n  H)$, i.e. the greatest common divisor of\n  the absolute values of the maximal minors of $H$ (see e.g.\\ \\cite{MR08}*{Lemma 5.1}). \n  \\end{rem}\n\nA morphism between tropical cycles\nis a locally affine linear map, with\nthe linear part induced by a map between the underlying lattices.\n A rational function $\\varphi$ on a tropical cycle $Y$ can be pulled back along a morphism $f:X\\rightarrow Y$\n  to the rational function $f^*(\\varphi) = \\varphi \\circ f$ on $X$. Also, we can push forward\n  subcycles $Z$ of $X$ to subcycles $f_*(Z)$ of $Y$ \\cite{AR07}*{Proposition\n  4.6 and Corollary 7.4}. \n\\vspace{0.3cm}\n\nWe refer the  reader  to \\cite{GKM07,Mi07,CJM10} for comprehensive background on moduli spaces of tropical curves and maps. A (marked, rational, abstract) tropical curve is a metric tree $\n\\Gamma $ without 2-valent vertices. Edges leading to 1-valent vertices have infinite length, are marked by the numbers $1,\\ldots,N$ and are called ends. The space of all marked tropical curves with\n$N$ ends is denoted $ \\mathcal{M}_{0,N} $. It can be embedded into $\\mathbb{R}^{\\binom{N}{2}-N}$ using the distance map. It follows from \\cite{SS04a}*{Theorem 3.4},\n\\cite{Mi07}*{Section 2}, or \\cite{GKM07}*{Theorem 3.7} that $\\mathcal{M}_{0,N}$ is a\ntropical variety which is even a fan. All top-dimensional cones have weight one. \nFor a subset $I\\subset\\left[N\\right]$ of cardinality $1<|I|<N-1$ define $v_I$\nto be the image under the distance map of a tree $\\Gamma$ with exactly one bounded\nedge of length one, the marked ends $i\\in I$ on one side and the\nends $i$ for $i\\notin I$ on the other. By abuse of notation, we often do not distinguish between a tree and its image under the distance map. We consider $\\mathbb{R}^{\\binom{N}{2}-N}$ not with its usual lattice, but with the lattice generated by the $v_I$.\nThe set of curves of a given combinatorial type is the interior of a\n$k$-dimensional cone of $\\mathcal{M}_{0,N}$, where $k$ denotes the number of bounded\nedges. The vectors $v_I$ generate the rays of $\\mathcal{M}_{0,N}$.\n\nWe denote by $\\TP= \\mathbb{R}\\cup {\\pm \\infty}$ the simplest model of tropical $\\mathbb{P}^1$.\nA tropical cover is a map $h:\\Gamma \\rightarrow \\TP$ where $\\Gamma$ is a tropical curve and $h$ is a continuous map which is integral affine on each edge. For each bounded edge, the stretching factor with which it is mapped to $\\mathbb{R}\\subset\\TP$ is called its weight.  \nEdges which are contracted have weight $0$. Two tropical covers are called isomorphic if they differ by a translation of the base $\\TP$.\n\nWe sometimes fix a reference orientation for the edges of $\\Gamma$. We then use the convention that (nonzero) weights are positive whenever the image of the tail in $\\mathbb{R}\\cup {\\pm \\infty}$ is smaller than the image of the head, and negative otherwise. At each vertex, a tropical cover satisfies the balancing condition, i.e.\\ the sum of the weights of incoming edges equals the sum of the weights of outgoing edges.\nThe degree $\\mathbf{x}$ of a tropical cover is the multiset of weights of its non-contracted ends, where we make the convention that all ends point away from their 1-valent vertex.\n\nWe fix  a degree $\\mathbf{x}$ and set $n=\\ell(\\mathbf{x})$. The space of isomorphism classes of tropical covers of degree $\\mathbf{x}$ and with additional $r$ marked ends which are contracted to points is called $ \\mathcal{M}_{0,r}(\\TP,\\mathbf{x}) $.\n\n\n\\begin{conv}\n\\label{conv}\nFor practical purposes in computations, it is useful to pick a distinguished representative for each equivalence class of tropical covers. By\nconvention, we introduce a contracted end labeled  $n+1$ and require that\nit is mapped to $0\\in \\TP$ by the map $h$. We say the end $n+1$ fixes a parameterization of $\\TP$.\n\\end{conv}\n\nBy \\cite{GKM07}*{Proposition 4.7}, $ \\mathcal{M}_{0,r}(\\TP,\\mathbf{x}) $ is a\n  tropical variety. It can be identified with $\\mathcal{M}_{0,N}$, where $N=n+r$, via the\n  map that forgets the map $h$.\nIf we later want to count tropical covers using the space \n$ \\mathcal{M}_{0,r}(\\TP,\\mathbf{x}) $ we overcount in the situation where we have multiple ends of the same weight, since we mark all the ends here and thus make them artificially distinguishable even if they have the same weight. For counting purposes, we thus have to divide by $|\\Aut(\\mathbf{x})|$. Since we are mostly interested in the situation where the degree $\\mathbf{x}$ is variable, this does not play an important role.\n\nTo compute intersection products in cells of\n  tropical moduli spaces, we work in the local coordinates of the appropriate cone of $\\mathcal{M}_{0,N}$, i.e.\\ in coordinates given by the lengths of each bounded edge.\n\nFor a subset $I\\subset\\left[N\\right]$, there\nis a forgetful map $\\ft_I:\\mathcal{M}_{0,N}\\longrightarrow \\mathcal{M}_{0,N-|I|}$ which\nmaps a tropical curve to the curve where we remove all ends with labels in $I$ (and possibly straighten $2$-valent vertices). Forgetful maps are\nmorphisms by \\cite{GKM07}*{Proposition 3.9}.\nThe forgetful map $\\ft:  \\mathcal{M}_{0,r}(\\TP,\\mathbf{x}) = \\mathcal{M}_{0,N} \\rightarrow \\mathcal{M}_{0,n}$ which forgets all contracted ends is most important.\n\n For each contracted end with label $i$  the evaluation map\n    $ \\ev_i : \\mathcal{M}_{0,r}(\\TP,\\mathbf{x}) \\rightarrow \\mathbb{R}\\subset\\TP $\n  assigns to a tropical curve the point in $\\mathbb{R}\\subset \\TP$ to which the end is contracted by $h$. It is shown in \\cite{GKM07}*{Proposition\n  4.8} that these maps are morphisms.\n\nFor each marked end $i$  the tropical Psi-class $\\Psi_i$, as a divisor, consists of all closed codimension one cones of $\\mathcal{M}_{0,N}$ parameterizing graphs  where the end $i$ is adjacent to a 4-valent vertex \\cite{Mi07}. All these cones come with weight one. It is also known how to intersect tropical Psi-classes \\cite{KM07}: every top-dimensional cone appearing in the intersection $\\Psi_1^{k_1}\\cdot \\ldots \\cdot \\Psi_N^{k_N}$ corresponds to a combinatorial type satisfying the following: if the ends $i_1,\\ldots,i_s$ are adjacent to a vertex $V$ then this vertex has valency $k_{i_1}+\\cdots+k_{i_s}+3$. We associate the weight $(k_{i_1}+\\cdots+k_{i_s})! \\cdot (k_{i_1}!\\ldots k_{i_s}!)^{-1}$ to the vertex $V$. The weight of a cone equals the product of these vertex weights for the corresponding combinatorial type.\n\nFor a subset $I\\subset \\{1,\\ldots,N\\}$ with $N-2 \\geq |I|\\geq 2$ we define the tropical boundary divisor $D_{I}$ to be the divisor cut out by the rational function $\\varphi_{I}$ that sends $v_{I}$ to one and all other vectors $v_{I'}\\neq v_I$ to zero \\cite{Rau08}*{Definition 2.4}.\n\n\\subsection{Combinatorial Set-Up}\n\\label{sec:csu}\n\nIn this section we introduce some basic combinatorial objects that play a role in the development of Hurwitz cycles. We think of Hurwitz loci as functions of the special ramification profiles. We define the domain of definition of such functions and outline the linear structure it has. \n\n\\begin{defn}\nFix a positive integer $n$. The lattice\n$$\n\\mathcal{H}=\\mathbb{Z}^n \\cap \\left\\{ \\sum_{i=1}^n x_i = 0 \\right\\}\n$$\nis the parameter space for double Hurwitz cycles. \nFor any proper subset $I\\subset \\{1, \\ldots, n\\}$, the hyperplane\n$$\nW_I= \\left\\{ \\sum_{i\\in I} x_i = 0 \\right\\}\n$$\nis called a \\textbf{wall}.  Any connected component of $\\mathcal{H}\\otimes \\mathbb{R} \\smallsetminus \\cup_I W_I$ is called a \\textbf{chamber}  (of polynomiality) and denoted by $\\mathfrak{c}$.\n\\end{defn}\n\nNext we introduce some notation for sets of decorated trees.\n\\begin{defn}\nLet $\\mathbf{x}\\in \\mathcal{H}$. We denote by $\\mathcal{T}_{r-k}(\\mathbf{x})$ the set of (non-metric) trees with $n$ ends labelled by the $x_i$'s and $r-k$ internal vertices of valence at least $3$.\nWe denote by $\\mathcal{T}^{\\mathfrak{c}}_{r-k}(\\mathbf{x})$ the set of tropical covers whose underlying tree is in $\\mathcal{T}_{r-k}(\\mathbf{x})$. Combinatorially, trees in $\\mathcal{T}^{\\mathfrak{c}}_{r-k}(\\mathbf{x})$ are further decorated with the following data:\n\\begin{enumerate}\n\t\\item A direction for the edges.\n\t\\item Positive weights for each edge in such a way that the balancing condition is satisfied at every vertex.\n\t\\item A total ordering of the vertices compatible with the direction of the edges.\n\\end{enumerate}\n\\end{defn}\n\nDecorations (1) and (2) can in fact be deduced uniquely from imposing the balancing condition, and depend on the chamber $\\mathfrak{c}$ that $\\mathbf{x}$ belongs to.\n\n\\begin{lemma}\n\\label{lem:flat}\nAny $\\Gamma$ in $\\mathcal{T}_{r-k}(\\mathbf{x})$ can be  ``lifted'' to a tropical cover as in Section \\ref{sec:tg}.\n\\end{lemma}\n\\begin{proof}\nFor most combinatorially versed people Lemma \\ref{lem:flat} is obvious. We include a brief sketch for the combinatorially impaired such as the majority of the authors of this paper (see \\cite{GKM07}*{Lemma 4.6} for a more honest discussion).\nAny tree with leaves labelled by an $n$-tuple of integers adding to $0$ can be given positive weights and orientations to all internal edges in a unique way by imposing the balancing condition:  give an arbitrary orientation to all edges in the tree. Cut any internal edge to obtain two smaller connected graphs. The cut edge can be assigned weight by requesting that each the sum of labels on each of the two connected graphs is $0$. Now the statement follows by induction. Wherever  one obtained a negative weight at an internal edge, switch the orientation of the edge with respect to the arbitrary one chosen earlier and switch the sign of the weight. Having given an orientation to any internal edge immediately allows to flatten the graph, concluding the proof. \n\\end{proof}\n\n\n\n\\begin{defn}\n\\label{mult}\nThe number of ways that a graph $\\Gamma$ can be flattened is  the number of vertex orderings for the graph that are compatible with the direction of the edges of the graph.  We call such number $m_\\mathfrak{c}(\\Gamma)$. If the chamber $\\mathfrak{c}$ is clear from the context, we also write $m(\\Gamma)$. \\end{defn}\n\nIn other words, the forgetful morphism $f:\\mathcal{T}^\\mathfrak{c}_{r-k}(\\mathbf{x})\\to \\mathcal{T}_{r-k}(\\mathbf{x})$ is surjective and $m_\\mathfrak{c}(\\Gamma)= |f^{-1}(\\Gamma)|$.\n\n\\begin{defn}\n\\label{phi}\nFor $\\Gamma \\in \\mathcal{T}^\\mathfrak{c}_{r-k}(\\mathbf{x}) $ (or in $\\mathcal{T}_{r-k}(\\mathbf{x})$ ) we define $\\varphi(\\Gamma)$ to be the product of the weights of all internal edges of $\\Gamma$.\n\\end{defn}\n\n\nBecause in each chamber $\\mathfrak{c}$ all edge weights are linear homogeneous polynomials in the $x_i$'s, the functions   $\\varphi(\\Gamma)$ are homogeneous polynomials in each chamber $\\mathfrak{c}$ of degree equal the number of internal edges of $\\Gamma$.\n\n\n\n\n\n\\section{Hurwitz Loci and Classes}\n\\label{sec:hl}\n\n\nIn this section we define Hurwitz classes and study their (piecewise) polynomial properties. We say that a family of Chow cycles $\\alpha(\\mathbf{x})$  in the moduli space of rational pointed stable curves is polynomial of degree $d$ and dimension $k$ if  $\\alpha(\\mathbf{x}) \\in Z_k(\\overline{M}_{0,n})[x_1, \\ldots, x_n]_d$. This is equivalent to \n$\\alpha$  having an expression as a combination of dimension $k$ boundary strata with coefficients polynomials in the $x_i$'s of degree $d$. \n\n\\begin{figure}[tb]\n\\input{T.pstex_t}\n\\caption{The union of strata we denote by $T^{r-k}\\subset \\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})$ paramereterizes  chains of $r-k$ rational curves, where the $i$-th component hosts the $i$-th (light) marked point, and the remaining $k$ points are distributed in all possible ways among the various twigs.}\n\\label{fig:T}\n\\end{figure}\n\n\n\\begin{defn}\nLet $\\mathbf{x}\\in \\mathcal{H}$. Consider the union  of boundary strata $$T^{r-k}\\subset \\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})$$ parameterizing  chains of $r-k$ projective lines, where the $i$-th component hosts the $i$-th (light) marked point, as illustrated in Figure \\ref{fig:T}. Referring to Figure \\ref{fig:natmorph} for the names of the natural morphisms, we define the $k$-dimensional Hurwitz cycle:\n\\begin{equation}\n\\mathbb{H}_k(\\mathbf{x}):=\\st_\\ast( \\br^\\ast(T^{r-k})) \\in Z_k(\\overline{M}_{0,n}).\n\\end{equation}\nSometimes we want to look at the $k$-dimensional Hurwitz locus in the appropriate space of maps. We make the definition:\n\\begin{equation}\n\\tilde\\mathbb{H}_k(\\mathbf{x}):= \\br^{-1}(T^{r-k}) \\subseteq \\overline{M}^\\sim_{0}(\\mathbf{x},t_1,\\ldots, t_{r-k}).\n\\end{equation}\n\\end{defn}\n\n\\begin{figure}[tb]\n\\centerline{\n\\xymatrix{\n\\tilde{\\mathbb{H}}_k(\\mathbf{x})   \\ar[d]  \\ar[r]   &  \\overline{M}^\\sim_0(\\mathbf{x}, t_1, \\ldots, t_j)  \\ar[d]^{\\br}\\ar[r]^{\\st}& \\overline{M}_{0,n}\\\\\n [T^{r-k}] \\ar[r]&  [\\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})\/S_k] & \\\\\n}}\n\\caption{The $k$-dimensional Hurwitz locus is the inverse image via the branch map of the stratum $T^{r-k}$.}\n\\label{fig:natmorph}\n\\end{figure}\n\n\\begin{rem}\nHurwitz loci were introduced in \\cite{gv:rvl}*{Section 4.4}. The only difference  in our definition is that we ``mark'' the branch points that we are fixing. The use of the more refined branch morphism to the Losev-Manin space gives us a more convenient expression of the locus in terms of the pull-back of a boundary stratum in $\\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})$. The next Lemma establishes the equivalence with Graber-Vakil's definition and is parallel to the definition we make on the tropical side.\n\\end{rem}\n\n\n\\begin{lemma}\nConsider the space of relative stable maps to a parameterized $\\mathbb{P}^1$, with   the natural stabilization morphism $\\st: \\overline{M}_{0,r-k}(\\mathbf{x})\\to \\overline{M}_{0,n}$.  Let $\\psi_i$ denote the first Chern class of the $i$-th cotangent line bundle, and $\\ev_i$ the $i$-the evaluation morphism. Then\n\\begin{equation}\n\\mathbb{H}_k(\\mathbf{x})=\\st_\\ast\\left(\\prod_{i=1}^{r-k} \\psi_i \\ev^\\ast_i([pt])\\right).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nInformally, the result is a combination of the three following facts: putting a $\\psi$ class is equivalent to requesting the mark to be a point of simple ramification for the map (\\cite{op:cc}); fixing one of the branch points gives the equivalence with the space of rubber-maps (\\cite{gv:rvl}*{Lemma 4.6}); via degeneration formula, any additional branch point that is fixed amounts to breaking the target into a chain with one more projective line. We expand on this third fact for the benefit of a reader who is not familiar with degeneration techniques.\n\nWe assume by induction that the class $\\prod_{i=1}^j \\ev_i^\\ast{[pt]}$ can be represented by the cycle of maps to a chain $T^j=\\cup_{i=1}^jT_i$ of $j$ projective lines with the $i$-th mark on the $i$-th line. Consider the trivial family $X=\\mathbb{C}\\times T^j$ and note that it comes with two horizontal sections: $\\mathbf{0_1}$ on $T_1$ and $\\mathbf{\\infty_j}$ on $T_j$. On $T_j$ consider the section $s(t)=1\/t$ that meets the section  $\\mathbf{\\infty_j}$ at $t=0$. Now let $X'$ be the blow up the point $(0,\\infty_j)$ in $X$ and consider the locus $\\ev_{j+1}^{\\ast}([s(t)])$ in the space of relative stable maps to $X'$ of degree $dF$ (where $F$ is the class of a fiber of the family and $d= \\sum x_i^+$ ), relative to the divisor $\\mathbf{x^+0_1}+\\mathbf{x^-\\infty_j}$. The degeneration formula (\\cites{lr:df, l:df})  asserts that it is equivalent to represent $F$ by a general fiber or by the central fiber of the family. In the first case we have $\\prod_{i=1}^{j+1} \\ev_i^\\ast{[pt]}$, in the second case we obtain the cycle of maps to the chain $T^{j+1}$ with the $i$-th mark on the $i$-th line.  \n\\end{proof}\n\n\nThe description of Hurwitz loci in terms of boundary strata in spaces of maps, together with an analysis of the multiplicities of the push-forwards to $\\overline{M}_{0,n}$ naturally leads to discover the (piecewise) polynomiality of the Hurwitz cycles.\n\n\\begin{thm}\n\\label{thm:poly}\nFor $\\mathbf{x}\\in \\mathfrak{c}$, $\\mathbb{H}_k(\\mathbf{x})$ is a homogeneous polynomial cycle of degree $n-3-k$.\n\\end{thm}\n\\begin{proof}\nTheorem \\ref{thm:poly} is a consequence of Lemmas \\ref{spec}, \\ref{flatten} and \\ref{coeff}, where we also give an explicit description of the (polynomial) multiplicity of each boundary stratum.\n\\end{proof}\n\nA simple observation that gives us a surprising amount of mileage, is that many boundary strata in $\\tilde\\bbH_k(\\mathbf{x})$ don't survive being pushed forward to $\\overline{M}_{0,n}$.\n\n\\begin{lemma}\n\\label{spec}\nLet  $\\mathbf{x}\\in \\mathfrak{c}$ and $\\tilde\\Delta$ an irreducible component of $\\tilde\\mathbb{H}_k(\\mathbf{x})$, corresponding to a stratum of maps to the rational chain $T=T_1\\cup\\ldots\\cup T_{r-k}$. Then $\\pi_\\ast(\\tilde\\Delta)\\not=0$ if and only if for every $i=1, \\ldots, r-k$ there is exactly one connected rational curve mapping non-trivially to $T_i$. \n\\end{lemma}\n\\begin{proof}\nLet $f:X\\to T$ correspond to a general point in $\\tilde\\Delta$.\nOver each $T_i$ there must be at least one connected component mapping non-trivially. By a trivial dimension count the push forward of $\\tilde\\Delta$ does not vanish precisely when the stabilization of $X$ is a rational curve with $r-k$ components and therefore there cannot be more than one non-trivial component over each $T_i$. \n\\end{proof}\n\nNext we observe that each boundary stratum of dimension $k$ in $\\overline{M}_{0,n}$ appears as the push forward of some stratum in $\\tilde\\bbH_k(\\mathbf{x})$.\n\n\\begin{lemma}\n\\label{flatten}\nLet   $\\mathbf{x} \\in \\mathfrak{c}$,  $\\Gamma\\in \\mathcal{T}_{r-k}(\\mathbf{x})$. Then there exists  an irreducible component $\\tilde\\Delta$ in $\\tilde\\mathbb{H}_k(\\mathbf{x})$ such that the stabilization of the source curve of the general element of $\\tilde\\Delta$ has dual graph $\\Gamma$.\n\\end{lemma}\n\\begin{proof}\nThis statement becomes  transparent after noting that boundary strata in $\\overline{M}^\\sim_{0}(\\mathbf{x},t_1,\\ldots, t_{r-k})$ are in bijective correspondence with ``tropical dual graphs''. Given a boundary stratum  $[S]$ whose general element is given by a map $f: X\\to T^{r-k}$: \n\\begin{itemize}\n\\item the chain $T^{r-k}$ is replaced by the interval $[0, r-k+1]$ with the $i$-th projective line corresponds to the point $i$ and $i$-th node corresponding to the segment $(i, i+1)$; the points $0$ and $r-k+1$ correspond to the relative points.\n\\item over point $i$ draw one vertex for each connected component of $f^{-1}(T_i)$; \n\\item over segment $(i,i+1)$ draw one edge for each node  of $X$ above the $i$-th node of $T^{r-k}$; the edge connects the appropriate vertices and is weighted with the ramification order at the node;\n\\item over $[0,1)$ and $(r-k,r-k+1]$ draw edges corresponding to the relative condition $\\mathbf{x}$.\n\\end{itemize}\nWe call the graph thus obtained the \\textit{pre-stable} tropical dual of $[S]$.  We stabilize the graph by forgetting all two-valent vertices to obtain what we call the tropical dual of $[S]$. \nThis procedure is illustrated in Figure \\ref{fig:flatten}  and clearly it can be inverted to identify a boundary stratum of maps given a tropical cover. With this translation Lemma \\ref{flatten} is equivalent to the purely combinatorial statement of Lemma \\ref{lem:flat}.\n\\end{proof}\n\n \n\n\\begin{figure}[tb]\n\\input{dual.pstex_t}\n\n\\caption{The construction of the tropical dual graph associated to a boundary stratum of relative maps. The $y_i^j$ are the degrees of the trivial covers of the $i$-th twig, and the $z_i^j$ the ramification orders over the $i$-th node. Note that the $z$'s on either side of a trivial cover are equal to the corresponding $y$ (e.g $x_1=y_1^1=z_1^1=y_2^1=z_2^1$). This makes it possible to erase the two valent vertices and have the $z_i^j$ become the weights of the edges of the tropical dual graph.}\n\\label{fig:flatten}\n\\end{figure}\n\n\\begin{lemma}\n\\label{coeff}\nLet   $\\mathbf{x} \\in \\mathfrak{c}$,  $\\Gamma\\in \\mathcal{T}_{r-k}(\\mathbf{x})$ and $\\Delta_{\\Gamma}$ the corresponding boundary stratum in $\\overline{M}_{0,n}$.  Let $m_\\mathfrak{c}(\\Gamma)$  and $\\varphi(\\Gamma)$ as in Definitions \\ref{mult}, \\ref{phi} and $f:\\mathcal{T}^\\mathfrak{c}_{r-k}(\\mathbf{x})\\to \\mathcal{T}_{r-k}(\\mathbf{x})$ be the natural forgetful morphism. Then\n\\begin{align}\n\\label{eq:pomult}\n\\mathbb{H}_k(\\mathbf{x}) & =    \\sum_{\\tilde\\Gamma \\in \\mathcal{T}^\\mathfrak{c}_{r-k}(\\mathbf{x})} \\varphi(\\tilde\\Gamma)\\prod_{v}(\\val(v)-2)\\ \\Delta_{f(\\tilde\\Gamma)} \\nonumber\\\\\n & =\\sum_{\\Gamma \\in  \\mathcal{T}_{r-k}(\\mathbf{x})}  m_\\mathfrak{c}(\\Gamma)\\varphi(\\Gamma) \\prod_{v}(\\val(v)-2)\\Delta_{\\Gamma}.\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nWe observed that  for each  boundary stratum  $\\Delta_{\\Gamma}$ in $\\overline{M}_{0,n}$ there are precisely $ m_\\mathfrak{c}(\\Gamma)$ boundary strata in $\\tilde\\bbH_k(\\mathbf{x})$ pushing forward to it. It remains to show that each such stratum $[S]$ pushes forward with multiplicity $ \\varphi(\\Gamma)\\prod_{v}(\\val(v)-2)$.  We obtain this multiplicity by adapting formula \\eqref{bound:multi} to the simple shape of the boundary strata we are observing. Over each $T_i$ in the chain there is only one connected component  of $X$ that maps non-trivially. Therefore we can replace the moduli space of possibly disconnected covers $\\mathcal{M}^\\bullet_i$ of \\eqref{bound:multi} by the unique moduli space of non-trivial connected covers to $T_i$, call it $\\mathcal{M}_i$, times an automorphism factor of $1\/y_i^j$ for every connected component of degree $y_i^j$ mapping  trivially to $T_i$; we denote by $\\triv_i$ the number of  trivial covers of the $i$-th component. Also the factors of $1\/|\\Aut(\\mathbf{z_i})|$ need to be replaced by the automorphisms of each sub-partition of $\\mathbf{z_i}$ corresponding to nodes on the same pair of curves on the two sides of the $i$-th node of $T^{r-k}$. But $X$ is a rational curve and hence all such sub-partitions have length one and trivially no automorphisms.  Hence \\eqref{bound:multi} becomes:\n\n\\begin{equation}\n\\label{eq:popo}\n[S]= \\frac{ \\prod_{i=1}^{r-k-1}\\prod_{j=1}^{r_i} z_i^j\n}{ \\prod_{i=1}^{r-k}\\prod_{j=1}^{{\\triv}_i} y_i^j}\n \\left[\\gl_\\ast\\left( \\prod_{i=1}^{r-k} \\mathcal{M}_i \\right)\\right].\n\\end{equation}\n\n\n\nEquation \\eqref{eq:pomult} is deduced from \\eqref{eq:popo} via the following observations:\n\\begin{enumerate}\n\\item There is a factor of  $y_i^j$ for each $2$-valent vertex  of the pre-stable tropical dual of $[S]$, and $y_i^j$ is equal to the weight of (either) edge on each side of the vertex. Therefore all $y_i^j$'s cancel with some of the $z$'s. The remaining multiplicity from the first part of \\eqref{eq:popo} is therefore the product of weights of all internal edges of the tropical dual graph: by definition this is   $\\varphi(\\Gamma)$.\n\\item Each $\\mathcal{M}_i$ is a moduli space of connected, rubber relative stable maps, with one simple transposition marked. By our discussion in Section \\ref{sec:mscm}, we know  $\\st$ maps  $\\mathcal{M}_i$ onto $\\overline{M}_{0,n}$ with degree $r={{r}\\choose{1}}$. If we call $v_i$ the vertex of the tropical dual graph corresponding to $\\mathcal{M}_i$, then in this case $n= \\val (v_i)$ and thus $r=\\val (v_i)-2$.\n\\end{enumerate}\n\\end{proof}\n\nTo obtain Theorem \\ref{thm:poly} it is now sufficient to remark that the weights of the edges of the tropical dual graph are linear homogeneous polynomials in the $x_i$'s and there are precisely $n-3-k$ internal edges in the tropical dual graph of any stratum  in $\\tilde\\bbH_k(\\mathbf{x})$ that pushes forward non-trivially to $\\overline{M}_{0,n}$.\n\nWe conclude this section by noting that our arguments do not apply exclusively to Hurwitz cycles, but to any cycle obtained by pull-pushing a boundary stratum from Losev-Manin to $\\overline{M}_{0,n}$. We thus obtain the following:\n\n\\begin{cor}\nLet $\\mathbf{x} \\in \\mathfrak{c}$  and $\\overline\\Delta$ be a union of boundary strata of dimension $k$ in $\\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})$. Then $\\st_\\ast \\br^\\ast(\\overline\\Delta)$ is a homogeneous polynomial class of degree $n-3-k$ in $\\mathfrak{c}$.\n\\end{cor}\n\n\n\n\n\n\n\\section{Tropical Hurwitz Loci}\n\n\\label{sec:thl}\n\n\\begin{defn}Let $\\mathbf{x}\\in \\mathcal{H}$ and $r=n-2$. We define the $k$-dimensional tropical Hurwitz cycle as\n$$\\mathbb{H}^{\\trop}_k(\\mathbf{x}):= \\ft_{\\ast}\\Big(\\Psi_{n+1}\\cdot \\prod_{i=n+2}^{n+r-k} \\big(\\Psi_i \\cdot \\ev_i^{\\ast}(p_i) \\big) \\cdot \\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x})\\Big) \\subset \\mathcal{M}_{0,n},$$\nwhere the $p_i$ are fixed arbitrary points in $\\mathbb{R}\\subset \\TP$. (The notions of tropical moduli spaces, intersection theory and Psi-classes we use here are as recalled in Section \\ref{sec:tg}.) We also define the Hurwitz locus:\n$$\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x}):= \\Psi_{n+1}\\cdot \\prod_{i=n+2}^{n+r-k} \\big(\\Psi_i \\cdot \\ev_i^{\\ast}(p_i) \\big) \\cdot \\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x}).$$\n\\end{defn}\n\n\n\nWe describe the (general) tropical curves parameterized by $\\mathbb{H}^{\\trop}_k(\\mathbf{x})\\subset \\mathcal{M}_{0,n}$: if the $p_i$'s and  $0$ are distinct then the evaluation pullbacks force the contracted ends $n+1,\\ldots,n+r-k$ to be adjacent to distinct vertices. Each Psi-class makes the corresponding contracted end adjacent to a $4$-valent vertex.  All contracted ends are forgotten in the push-forward. Thus the top-dimensional cells of  $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ consist of  tropical curves that admit a cover of degree $\\mathbf{x}$ to $\\TP$ such that $r-k$ of its $r$ $3$-valent vertices are mapped to the points $p_i$ and $0$.\n\n\\begin{rem}\nIf $k=0$, all $r$ $3$-valent vertices are fixed. \nSince the evaluation maps at the contracted ends in the intersection of Psi-classes coincides with the tropical branch map (see \\cite{CJM10}*{Definition 5.14}),\neach point in the $0$-dimensional cycle $\\mathbb{H}^{\\trop}_0(\\mathbf{x})$ comes with a multiplicity which equals the multiplicity of the tropical branch map. The degree of the $0$-dimensional cycle $\\mathbb{H}^{\\trop}_0(\\mathbf{x})$ thus equals $|\\Aut(\\mathbf{x})|$ times the tropical Hurwitz number $H^0(\\mathbf{x})$ as  defined in \\cite{CJM10}. \n\\end{rem}\n\n\\begin{rem}\\label{rem:equivalent}\nThe cycle $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ depends on the choice of the $p_i$.\n By \\cite{AR07} however, different choices of $p_i$ yield rationally equivalent cycles. \n\\end{rem}\n\n\\begin{lemma}\\label{lem-evproduct}\n Let $\\tilde{\\Gamma}$ be a combinatorial type of a cover in  $\\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x})$ with $r-k$ vertices of valence at least $4$ and with each contracted end $n+1,\\ldots,n+r-k$ adjacent to one of the $r-k$ vertices. Then the evaluation map $\\ev_{n+2}\\times \\ldots \\times \\ev_{n+r-k}$ in local coordinates of the cone of $\\tilde{\\Gamma}$ is a square matrix such that the absolute value of its determinant equals the product of the weights of the bounded edges of $\\tilde{\\Gamma}$.\n\\end{lemma}\n\\begin{proof}\n Inductively we can order the vertices which are not adjacent to $n+1$ and the bounded edges of $\\tilde{\\Gamma}$ so that going from end $n+1$ to another contracted end\n only bounded edges with a smaller index are traversed\n\n (see also \\cite{CJM10}*{Lemma 5.26}). Then the evaluation map in local coordinates is a square triangular matrix such that the diagonal entries are the weights of the bounded edges of $\\tilde{\\Gamma}$.\n\\end{proof}\n\n\n\n\n\\begin{defn}\\label{def:weights} Let $\\mathbf{x}\\in \\mathcal{H}$, and\n $\\Gamma$  a combinatorial type of a marked tropical curve in $\\mathcal{M}_{0,n}$. Associate to each end $i$ of $\\Gamma$ the weight $x_i$. The weights of the other edges are determined by the balancing condition (see Lemma \\ref{lem:flat}). Orient the edges of $\\Gamma$ so that all weights are positive. The orientation of the edges induces a partial ordering of the vertices of $\\Gamma$.\nFor any choice of $k$ vertices $V_1,\\ldots,V_k$ of $\\Gamma$ we denote by $m_{V_1,\\ldots,V_k}(\\Gamma)$ the number of ways to totally order the vertices different from $V_1,\\ldots,V_k$ of $\\Gamma$ respecting the partial order induced by the orientation (see also \\cite{CJMwc}*{Definition 2.1 and 2.15}). (If $k=0$, this agrees with $m(\\Gamma)$ from Definition \\ref{mult}.) We call the vertices $V_1,\\ldots,V_k$ which are not ordered the \\emph{moving vertices}.\n\nFor a fixed choice of $k$ moving vertices, let $\\Gamma'$ be a graph obtained from $\\Gamma$ by shrinking edges in such a way that all moving vertices are identified with a non-moving vertex, and no pair of non-moving vertices is identified. We denote by $w_{V_1,\\ldots,V_k}(\\Gamma)$ the greatest common divisor, over all possibilities to choose $\\Gamma'$, of the products of the weights of the bounded edges of $\\Gamma'$.\n\\end{defn}\n\n\\begin{defn}\n For a combinatorial type $\\tilde{\\Gamma}$ of tropical covers and a vertex $V$ of $\\Gamma$, we define the \\emph{moving vector} of $V$ to be the vector in $\\mathcal{M}_{0,N}$ that we have to add to a tropical cover of type $\\Gamma$ in order to move $h(V)$ to the right by an interval of legths $\\prod w(e)$ where the product goes over all bounded edges $e$ adjacent to $V$ and $w(e)$ denotes their weight.\nThe moving vector  corresponds to a tree with nonzero lengths for the bounded edges of $\\Gamma$ adjacent to $V$: each edge $e$ pointing out of $V$ has  length $-\\prod_{e'\\neq e}w(e')$, each edge pointing into $V$ has length $\\prod_{e'\\neq e}w(e')$; in both cases the product is over bounded edges $e'$ adjacent to $V$ and not equal to $e$ (see \\cite{GMO}*{Definition 5.5}).\nWe also call the image of a moving vector in $\\mathcal{M}_{0,n}$ under the forgetful map $\\ft$ a moving vector.\n\\end{defn}\n\n\n\n\\begin{lemma}\\label{lem:cellsweights} Let $\\Gamma$ be a combinatorial type of a top-dimensional cone of $\\mathcal{M}_{0,n}$.\nIf the points $p_i$ are chosen generically (i.e.\\ pairwise distinct and different from $0$), there are $\\sum m_{V_1,\\ldots,V_k}(\\Gamma)$ top-dimensional cells of  $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ contained in the cone of $\\Gamma$, where the sum goes over all $\\binom{r}{k}$ choices of $k$ moving vertices $V_1,\\ldots,V_k$ of $\\Gamma$. For a fixed choice of $k$ moving vertices, each of the $m_{V_1,\\ldots,V_k}(\\Gamma)$ cells corresponding to this choice has weight $w_{V_1,\\ldots,V_k}(\\Gamma)$. Moreover, all cells corresponding to a fixed choice of $k$ moving vertices are parallel, i.e.\\ their linear part is the same.\n\\end{lemma}\n\\begin{proof}\n A cell of $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ maps to the cone of $\\Gamma$ in $\\mathcal{M}_{0,n}$ if we obtain the combinatorial type $\\Gamma$ after forgetting the ends $n+1,\\ldots,n+r-k$. \nThus to recover such a cell from the graph $\\Gamma$ one must to pick $r-k$ vertices to which we attach the marked ends $n+1,\\ldots,n+r-k$ in such a way that we get a cover mapping the end $i$ to the point $p_i$ for $i=n+2,\\ldots,n+r-k$ and end $n+1$ to $0$. One can do this if and only if the orientation of the edges \nrespects the order of the image points $p_i$ of the $r-k$ vertices to which the mark ends are attached. \n\nThe graphs $\\Gamma'$ in Definition \\ref{def:weights}\n correspond to faces of the boundary of the cone of $\\Gamma$ that each cell corresponding to $V_1,\\ldots,V_k$ intersects in a point. \nThus two cells of $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ corresponding to different choices of moving vertices are pushed forward to different cells of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ by $\\ft$ --- their images under $\\ft_\\ast$ intersect different boundary cones of $\\Gamma$ in points. But also two cells corresponding to the same choice of moving vertices are pushed forward to different cells: they intersect the same boundary cones of $\\Gamma$, but they do so in different points as the coordinates of the intersection points are given by the lengths of the bounded edges of each $\\Gamma'$, and these lengths depend on the choice of which of the $r-k$ non-moving vertices is mapped to which of the $p_i$.\n\nIt follows that there are $\\sum m_{V_1,\\ldots,V_k}(\\Gamma)$ top-dimensional cells of  $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ contained in the cone of $\\Gamma$.\n\nWe now compute the weight of one such cell. We start by computing the weight of a cell in $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$. Each cell of $\\prod_{i=n+1}^{n+r-k}\\Psi_i$ corresponds to a combinatorial type where the ends $n+1,\\ldots,n+r-k$ are adjacent to different $4$-valent vertices. The weight of such a cone in the intersection $\\prod_{i=n+1}^{n+r-k}\\Psi_i$ equals one by \\cite{KM07}. \nBy Remark \\ref{rem-intersect} the intersection multiplicity from the evaluation pullbacks is the gcd of the absolute values of the maximal minors of the matrix of the evaluation map in local coordinates. \nWe evaluate each of the $r-k-1$ non-moving vertices different from $n+1$ (which is sent to $0$ by convention). A maximal minor corresponds to a choice of $r-k-1$ bounded edges: we contract the remaining bounded edges and get a graph $\\Gamma'$ with $r-k$ vertices.\n\n If some of the non-moving vertices  in $\\Gamma'$  are identified, then the minor has some identical rows (resp.\\ a zero row) and is thus zero. \n\n All the $r-k$ non-moving vertices must thus remain separate; each of the moving vertices is identified with some non-moving vertex, as in the definition of $w_{V_1,\\ldots,V_k}(\\Gamma)$. \nFinally Lemma \\ref{lem-evproduct} implies that the absolute value of the maximal minor corresponding to a possible choice of $\\Gamma'$ equals the product of the weights of the bounded edges of $\\Gamma'$. \n\n The weight of a cell of $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ equals $w_{V_1,\\ldots,V_k}(\\Gamma)$. The push forward maps cells of this intersection to cells of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ one by one, and the forgetful map has index one since it is just a projection. Thus the claim about the weight of cells of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ follows.\n\nFor any cell  $\\tilde{\\Gamma}\\in \\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ where $V_1,\\ldots,V_k$ are the moving vertices, the linear part of the affine hull of the cell $\\alpha$ is  spanned by the moving vectors of the vertices $V_1,\\ldots,V_k$ in $\\mathcal{M}_{0,N}$. Thus all cells of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ in the cone of $\\Gamma$ corresponding to a fixed choice of $k$ moving vertices are parallel.\n\\end{proof}\n\nLemma \\ref{lem:cellsweights} implies that a (top-dimensional) cell of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ is specified by a (3-valent) graph $\\Gamma$, a choice of $k$ moving vertices and an ordering of the remaining vertices that respects the edge orientations given by $\\mathbf{x}$.\nWe call this data  the \\emph{combinatorial type} of a cell of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$.\n\n\n\n\\begin{rem}\n $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ is not necessarily a simplicial polyhedral complex. Figure \\ref{fig:notsimplic} shows a combinatorial type corresponding to a two-dimensional cell. The moving vertices are labelled with an $M$. This cell has four vertices, since the moving vertices can merge with 1 and 3, or 1 and 2, or 2 and 3, or both with 2.\n\n\\begin{figure}[tb]\n\\input{notsimplic.pstex_t}\n\\caption{The combinatorial type of a twodimensional cell with four vertices.}\n\\label{fig:notsimplic}\n\\end{figure}\n\n\n\\end{rem}\n\n\n\nSince we define $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ as a tropical intersection product, it is of course a tropical cycle itself and thus in particular balanced around each cell of codimension one. We find it useful to have a concrete local description of the balancing of tropical Hurwitz cycles. We conclude this section by discussing this. \n\n\n\n\n\nThe Hurwitz cycle $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ has the structure of a weighted marked polyhedral complex (see \\cite{GMO}*{Definition 5.1}): let $\\sigma$ denote a top-dimensional cell of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ and $\\tilde{\\sigma}$ its unique preimage in $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$. Let $\\tilde{\\Gamma}$ be the combinatorial type of the cell $\\tilde{\\sigma}$. \nWe denote by $\\Gamma$ the image of $\\tilde{\\Gamma}$ under the forgetful map. A bounded edge of $\\Gamma$ is called \\emph{a fixed edge}, if the two end vertices of its preimage in $\\tilde{\\Gamma}$ are adjacent to two contracted ends (i.e.\\ their image is fixed by the evaluation pullback). We call a bounded edge \\emph{moveable}, if neither of the two end vertices of its preimage is adjacent to a contracted end.\nWe define the weight factor $\\omega'(\\sigma)$ to be the product of the weights of all fixed edges divided by the product of the weights of moveable edges.\nThe $k$ moving vectors of the moving vertices are the marked vectors of $\\sigma$. \n\nWe equipped the polyhedral complex underlying $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ with two a priori different weight functions for its top-dimensional cells: one weight function is defined by the tropical intersection product, the other by the structure of a marked polyhedral complex. It can be shown that these two structures agree. Assuming this, we now check the balancing condition (as in \\cite{GMO}*{Lemma 5.2}) locally around a codimension one face.\n\n\n\n\nA cell $\\tau$ of codimension one of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ \nparameterizes trees with exactly one $4$-valent vertex. There are two types of cells of codimension one: those where two moving vertices have merged into a $4$-valent vertex, and those where a moving vertex has merged with a non-moving vertex.\n\nWhen two moving vertices have merged, there are three top-dimensional neighbors of $\\tau$, corresponding to the three resolutions of the $4$-valent vertex.\n\n\nThe three neighboring cones  share $k-2$ marked vectors: the remaining two marked vectors  (for each cone) correspond to the moving vectors of the  new $3$-valent vertices obtained resolving the $4$-valent vertex. \nIn all three cases we can replace one of the two new moving vectors by the moving vector of the $4$-valent vertex of $\\tau$. Now the hypothesis of Proposition 5.2 in \\cite{GMO} apply, and we can show just as in Proposition 5.8 of \\cite{GMO} that the balancing condition at $\\tau$ is equivalent to  the appropriate weighted sum of the three other new moving vectors equal to the moving vector of the $4$-valent vertex. \n\nAssume now that a moving vertex is merged into a non-moving vertex to form the $4$-valent vertex $V$ of $\\tau$.\nWithout restriction, assume that one bounded edge of weight $a$ points into $V$ and three bounded edges of weights $b$, $c$ and $d$ point out of $V$.\n Then there are six top-dimensional neighbors of $\\tau$, illustrated in Figure \\ref{fig:neighbours}.\n\n\nAll but one of the marked vectors coincide for the six top-dimensional neighbors, they only differ by the moving vector of the new vertex. The moving vertex is indicated with an $M$.\nThe weight factors of the six neighboring cells differ only by two of the four bounded edges adjacent to $V$ which are fixed edges according to where the new moving vertex is.\n\n\\begin{figure}[tb]\n\\input{neighbours.pstex_t}\n\\caption{The six neighbors of a cell of codimension one where a moving vertex is merged with a non-moving vertex.}\n\\label{fig:neighbours}\n\\end{figure}\n\n\nWe apply Proposition 5.2 of \\cite{GMO} and deduce that the balancing condition around $\\tau$ is equivalent to the equation: \n\\begin{align*}\n&ab\\cdot (cd\\cdot v_{ab}- (a-b)d \\cdot v_c - (a-b) c \\cdot v_d)\\\\\n&-  cd\\cdot (-ab\\cdot v_{ab}+ (a-b)b \\cdot v_a - (a-b) a \\cdot v_b)\\\\\n&+ac\\cdot (bd\\cdot v_{ac}- (a-c)d \\cdot v_b - (a-c) b \\cdot v_d)\\\\\n&-bd\\cdot (-ac\\cdot v_{ac}+ (a-c)c \\cdot v_a - (a-c) a \\cdot v_c)\\\\\n&+ad\\cdot (bc\\cdot v_{ad}- (a-d)c \\cdot v_b - (a-d) b \\cdot v_c)\\\\\n&-bc\\cdot (-ad\\cdot v_{ad}+ (a-d)d \\cdot v_a - (a-d) a \\cdot v_d)\n\\\\&= 2abcd\\cdot (v_{ab}+v_{ac}+v_{bc}-(v_a+v_b+v_c+v_{abc}))=0,\n\\end{align*}\nwhere we use the equation $a=b+c+d$ and relations among the vectors $v_I$ as in \\cite{KM07}*{Lemma 2.6} (the vector $v_a$ here stands for the tree with all ends that we can reach from $V$ via $a$ on one side).\n\n\\subsection{Extended Example: Hurwitz Curves}\nWe study the tropical structure of the Hurwitz curve $\\mathbb{H}^{\\trop}_1(\\mathbf{x})$. Top dimensional cells (segments) parameterize graphs with one moving vertex.\nEach codimension one cell --- i.e.\\ vertex --- of the Hurwitz curve\ncorresponds to curves where the moving vertex is merged with another (fixed) vertex, and has six top-dimensional cells incident to it. For any $\\mathbf{x}$, the Hurwitz curve is a six-valent tropical curve in  $\\mathcal{M}_{0,n}$. We  distinguish several types of edges and compute their weights $w_{V_1}(\\Gamma)$. \n\n\n\n\n\\begin{figure}[tb]\n\\input{easyend.pstex_t}\n\\caption{The combinatorial type of a constant end of the Hurwitz curve.}\n\\label{fig:easyend}\n\\end{figure}\n\n\\emph{Constant Ends} (Figure \\ref{fig:easyend}) of the Hurwitz curve are edges corresponding to a combinatorial type where the moving vertex is adjacent to two ends $i$ and $j$. The moving vector equals $\\pm v_{ij}$. \nA constant end is incident to a unique vertex, corresponding to shrinking the only bounded edge adjacent to the moving vertex. \nThe weight of a constant end equals the product of all bounded edge weights except for the bounded edge adjacent to the moving vertex. \n\n\n\\begin{figure}[tb]\n\\input{complicatedend.pstex_t}\n\\caption{The combinatorial type of a linear end of the Hurwitz curve.}\n\\label{fig:complicatedend}\n\\end{figure}\n\n\\emph{Linear Ends} (Figure \\ref{fig:complicatedend}):\n the moving vertex is adjacent to two bounded edges $e_1$ and $e_2$ of the same direction and one end $i$, necessarily of opposite direction. The moving vector of a linear end equals $\\pm( w(e_2) v_{I}+w(e_1) v_{I\\cup \\{i\\}})$, where $I$ denotes the set of ends separated from $i$ via $e_1$.\n A linear end is incident to only one $0$-dimensional cell, corresponding to the graph where the moving vertex coincides with the vertex of the shorter of the two edges (the end vertex of $e_1$ in the picture). To obtain possible graphs $\\Gamma'$, we can shrink either $e_1$ or $e_2$. Thus the weight of a linear end equals $\\prod_{e\\neq e_1,e_2} w(e)\\cdot \\mbox{gcd}(w(e_1),w(e_2))$ where the product goes over all bounded edges $e$ except $e_1$ and $e_2$.\n\n\n\\begin{figure}[tb]\n\\input{easybounded.pstex_t}\n\\caption{The combinatorial type of a linear edge of the Hurwitz curve.}\n\\label{fig:easybounded}\n\\end{figure}\n\n\n\\emph{Linear Edges} (Figure \\ref{fig:easybounded}) are edges corresponding to a combinatorial type where the moving vertex is adjacent to two bounded edges $e_1$ and $e_2$ of opposite direction, and to an end $i$. The moving vector of an linear edge equals $\\pm( w(e_2) v_{I}- w(e_1) v_{I\\cup \\{i\\}})$ where $I$ denotes the set of ends separated from $i$ via $e_1$. We reach the two vertices of a linear edge by shrinking either $e_1$ or $e_2$. The weight of a linear edge is exactly as for a linear end: $\\prod_{e\\neq e_1,e_2} w(e)\\cdot \\mbox{gcd}(w(e_1),w(e_2))$.\n\n\\begin{figure}[tb]\n\\input{complicatedbounded.pstex_t}\n\\caption{The combinatorial type of a quadratic edge of the Hurwitz curve.}\n\\label{fig:complicatedbounded}\n\\end{figure}\n\n\n\\emph{Quadratic Edges} (Figure \\ref{fig:complicatedbounded}) are edges corresponding to a combinatorial type where the moving vertex is adjacent to three bounded edges $e_1$, $e_2$ and $e_3$. Without restriction we can assume that $e_1$ and $e_2$ are of the same direction, and $e_3$ is of opposite direction. We can also assume that the non-moving vertices are ordered such that the other end vertex of $e_1$ is between the end vertex of $e_2$ and the moving vertex. A quadratic edge connects the two vertices where we shrink $e_3$ resp.\\ $e_1$.\n The moving vector of a quadratic edge equals $\\pm( w(e_2)\\cdot w(e_3) v_{I}+ w(e_1)w(e_3) v_{J}- w(e_1)w(e_2) v_{(I\\cup J)^c})$ where $I$ denotes the set of ends that we can reach from the moving vertex via $e_1$ and $J$ denotes the set of ends that we can reach via $e_2$. \nWe can shrink each of the three adjacent edges to obtain a possible graph $\\Gamma'$ to compute the weight. Thus the weight of a complicated bounded edge equals $\\prod_{e\\neq e_1,e_2,e_3} w(e)\\cdot \\mbox{gcd}(w(e_1)w(e_2),w(e_1)w(e_3),w(e_2)w(e_3))$.\n\n\\begin{rem}\nThe weight of a one-dimensional cell $H$ in the Hurwitz curve is a function of the weights of the bounded edges of a curve $\\tilde{\\Gamma}$ parameterized by the cell. If we artificially define the weight of ends of  $\\tilde{\\Gamma}$ to be $1$ (these are not the right tropical weights!) then we can describe the weight of $H$ in a uniform way as the product of weights of all edges of  $\\tilde{\\Gamma}$ that are not incident to the moving vertex times the $gcd$ of the three products of  weights of pairs of edges incident to the moving vertex. Given that  weights of edges  of  $\\tilde{\\Gamma}$ are either $1$ or linear polynomials, the degree of the above $gcd$ can range from $0$ to $2$, thus explaining the names we chose above.\\end{rem}\n\n\n\n\n\\begin{example}\\label{ex:m05hurwitzcurve}\n We compute the Hurwitz curve $\\mathbb{H}^{\\trop}_1(\\mathbf{x})$ for $\\mathbf{x}=(x_1,\n\\ldots,x_5)$ with $x_1, x_2>0$, $x_3,x_4,x_5<0$, $x_1>|x_i+x_j|$ for all $i\\not=j\\in\\{3,4,5\\}$ and $x_2<-x_i$ for all $i=3,\\ldots,5$ in $\\mathcal{M}_{0,5}$.\nWe start with the cells in the cone spanned by $v_{12}$ and $v_{34}$ in $\\mathcal{M}_{0,5}$. To fix a combinatorial type of covers such that the corresponding cell of $\\mathbb{H}^{\\trop}_1(\\mathbf{x})$ lives in this cone, we first have to choose a moving vertex among the three vertices of the tree with ends $1$ and $2$ coming together and ends $3$ and $4$ coming together. For each of these three choices, there is one ordering of the remaining vertices which is compatible with the orientation of the edges (see Figure \\ref{fig:m05}). We thus have three cells of $\\mathbb{H}^{\\trop}_1(\\mathbf{x})$ in this cone, two of these cells are constant ends, the other one is an linear edge. Figure \\ref{fig:m052} shows the cone and the three cells.\nBy symmetry, the cones spanned by $v_{12}$ and $v_{35}$ resp.\\ by $v_{12}$ and $v_{45}$ look analogous. Similar arguments also show that all remaining cones except the cones spanned by $v_{23}$ and $v_{45}$, resp.\\  $v_{24}$ and $v_{35}$, resp.\\  $v_{25}$ and $v_{35}$, look alike.\n\n\\end{example}\n\\begin{figure}[tb]\n\\input{m05.pstex_t}\n\\caption{The combinatorial types of the Hurwitz curve in the cone of $\\mathcal{M}_{0,5}$ spanned by $v_{12}$ and $v_{34}$.}\n\\label{fig:m05}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\input{m052.pstex_t}\n\\caption{The Hurwitz curve in the cone of $\\mathcal{M}_{0,5}$ spanned by $v_{12}$ and $v_{34}$.  We require the vertex $6$ to be mapped to $0$ as usual and $7$ to $1$.}\n\\label{fig:m052}\n\\end{figure}\n\nIn the cone spanned by $v_{23}$ and $v_{45}$, we have two constant ends when the moving vertex is adjacent to two ends. If the third vertex is moving, there are two orderings of the remaining vertices compatible with the orientation of the edges (see Figure \\ref{fig:m053}). Altogether, we get two constant and two linear ends as depicted in Figure \\ref{fig:m054}.\n\n\\begin{figure}[tb]\n\\input{m053.pstex_t}\n\\caption{The combinatorial types of the Hurwitz curve in the cone of $\\mathcal{M}_{0,5}$ spanned by $v_{23}$ and $v_{45}$.}\n\\label{fig:m053}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\input{m054.pstex_t}\n\\caption{The Hurwitz curve in the cone of $\\mathcal{M}_{0,5}$ spanned by $v_{23}$ and $v_{45}$.}\n\\label{fig:m054}\n\\end{figure}\n\n\n\n\n\n\\section{Tropical-Classical Correspondence}\n\\label{sec:tcc}\n\nOur work in Sections \\ref{sec:hl} and \\ref{sec:thl} has highlighted a combinatorial correspondence between  classical and tropical Hurwitz cycles. In this section we make a precise statement of such a correspondence, and illustrate it in the example of one dimensional cycles. A satisfactory correspondence should also encode the polynomial multiplicities of strata. In Section \\ref{sec:int} we interpret the (classical) multiplicity of a stratum in the Hurwitz cycle as an intersection multiplicity of the corresponding tropical face with the  $k$-dimensional skeleton of $\\mathcal{M}_{0,n}$.\n\\begin{co}\n\\label{co:tropclas}\nThere is a natural bijection between $i$-dimensional faces of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ and connected components in  $\\tilde\\mathbb{H}_k(\\mathbf{x})$ of the inverse image via $\\st$ of irreducible  strata in $\\mathbb{H}_k(\\mathbf{x})$ of dimension $k-i$. Further, incidence of faces on the tropical side corresponds to intersection of strata on the classical side.  \n\\end{co}\n\nThe key ingredient here is the correspondence between tropical graphs and boundary strata of moduli spaces of relative stable maps outlined in Lemma \\ref{flatten}. The subtlety to observe is that a cell of the tropical Hurwitz cycle is sensitive to the type of the directed graph parameterized and to the ordering of the fixed vertices, but not to the ordering of moving vertices amongst themselves or with respect to the fixed ones. So two general points  $P_1,P_2$ in the same $i$-dimensional tropical cell may correspond to graphs where two adjacent vertices (at least one of which is moving) have switched order, and hence to different $(k-i)$-dimensional boundary strata $\\tilde{\\Delta}_1, \\tilde{\\Delta}_2$ of relative stable maps. The segment joining $P_1$ and $P_2$ contains a point $P$ where the two incriminated vertices map to the same image point: this corresponds to a $(k-i+1)$-dimensional boundary stratum $\\Delta$ that contains $\\tilde{\\Delta}_1$ and $\\tilde{\\Delta}_2$ as specializations. The stratum $\\Delta$ does not belong to $\\tilde{\\mathbb{H}}_k(\\mathbf{x})$, but it does belong to the inverse image via $\\st$ of the Hurwitz cycle.  We feel that describing this phenomenon in full generality would only mire us in notational confusion, so we choose  to illustrate it in one very specific example.\n\n\n\\subsection{Hurwitz Curves Continued}\n\nConsider the one dimensional cell labelled $I$ in Figure \\ref{fig:m054}, and the corresponding graphs parameterized by points in such cell (these are illustrated in Figure \\ref{fig:m053}). Let $\\ell$ be a coordinate for this cell, corresponding to the length of the segment joining vertices $6$ and $M$. For $\\ell < -\\frac{1}{x_4+x_5}$ (resp. $\\ell > -\\frac{1}{x_4+x_5}$) any point of $I$ is the tropical dual graph to the stratum $\\tilde{\\Delta}_1$ (resp. $\\tilde{\\Delta}_2$) as depicted in Figure \\ref{fig:contract}. The point $\\ell = -\\frac{1}{x_4+x_5}$ correspond to the stratum $\\tilde{\\Delta}$, which is a ${\\mathbb{P}^1}$ connecting $\\tilde{\\Delta}_1$ and  $\\tilde{\\Delta}_2$ in $\\st^{-1}({\\mathbb{H}_1}(\\mathbf{x}))$.\n\n\\begin{figure}[tb]\n\\input{strata.pstex_t}\n\\caption{Strata of relative stable maps in $\\st^{-1}({\\mathbb{H}_1}(\\mathbf{x}))$.}\n\\label{fig:contract}\n\\end{figure}\n\n\\subsection{The Hurwitz cycle \nintersecting the codimension $k$-skeleton of $\\mathcal{M}_{0,n}$}\n\n\\label{sec:int}\n\n\n\nIn this section we realise the (classical) multiplicities of the strata in the Hurwitz cycle as intersection numbers of the tropical Hurwitz cycle with the codimension  $k$-skeleton of $\\mathcal{M}_{0,n}$. To do so we must  view each codimension $k$ cell as a part of an intersection product of divisors. \nWe recall the boundary divisors $D_I$  that ``play well'' in the intersection theory of  $ \\mathcal{M}_{0,n}$ are defined as divisors of appropriate rational functions \\cite{Rau08}*{Definition 2.4}. Each $D_I$ is a linear combination of codimension one cells of $ \\mathcal{M}_{0,n}$ with appropriate weights. In the following lemma we describe some intersection products of these boundary divisors in term of the tropical curves they parameterize.\n\\begin{lemma}\\label{lem:divintersect}\nThe intersection of the tropical boundary divisors $ D_{12}\\cdot D_{123}\\cdot  \\ldots \\cdot D_{1...j}$ for some $j\\geq 2$ in $\\mathcal{M}_{0,m}$ consists of all cones corresponding to a type with the ends $1,\\ldots,j$ at\n\\begin{itemize}\n \\item  a $j+2$-valent vertex and only $3$-valent vertices otherwise with weight one,\n\\item  a $j+1$-valent vertex adjacent to a $4$-valent vertex and only $3$-valent vertices otherwise with weight $-1$.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nWe show this by induction. The induction beginning is \\cite{Rau08}*{Lemma 2.5}. For the induction step, assume that the statement is true for $j-1$. \nThe codimension $j$ cones in  $ D_{12}\\cdot D_{123}\\cdot  \\ldots \\cdot D_{1...j-1}$ then consist of all cones corresponding to a type with the ends $1,\\ldots,j-1$ at\n\\begin{itemize}\n \\item a $j+2$-valent vertex and only $3$-valent vertices otherwise,\n\\item  a $j+1$-valent vertex and one $4$-valent vertex,\n\\item  a $j$-valent vertex adjacent to a $5$-valent vertex,\n\\item  a $j$-valent vertex adjacent to a $4$-valent vertex and one other $4$-valent vertex.\n\\end{itemize}\nTo obtain the coefficients of such cones in the product   $ D_{12}\\cdot D_{123}\\cdot  \\ldots \\cdot D_{1...j}$ we must compute the intersection with $\\varphi_{1...j}$ around each of these cones.\n\nIn the first case, the neighbors of  $ D_{12}\\cdot D_{123}\\cdot  \\ldots \\cdot D_{1...j-1}$ are the resolutions of the $j+2$-valent vertex in a $j+1$-valent vertex with $1,\\ldots,j-1$ adjacent. Each such neighbor is spanned by a vector corresponding to this resolution. Since only the vector $v_{1...j}$ is mapped to one by $\\varphi_{1...j}$, we get a contribution of one for the weight of the codimension one cone only if also $j$ is adjacent to the $j+2$-valent vertex. The weight then equals one.\n\nIn the second case, we can resolve the $j+1$-valent vertex, but none of the resolutions is spanned by the vector $v_{1..j}$. We can also resolve the $4$-valent vertex. Again, none of the resolutions is spanned by the vector $v_{1...j}$, however, this vector is contained in the codimension one cone if also $j$ is adjacent to the $j+1$-valent vertex. If in addition the $4$-valent vertex is adjacent to the $j+1$-valent vertex, the sum of the vectors spanning its three resolutions contains $v_{1...j}$ as a summand: if we denote the four edges adjacent to the $4$-valent vertex by $e_1,\\ldots,e_4$ and assume that the subset of ends that can be reached via $e_i$ from the $4$-valent vertex is $A_i$, then the three resolutions are spanned by $v_{A_1\\cup A_2}$, $v_{A_1\\cup A_3}$ and $v_{A_2\\cup A_3}$ and their sum satisfies $v_{A_1\\cup A_2}+v_{A_1\\cup A_3}+v_{A_2\\cup A_3}=v_{A_1}+v_{A_2}+v_{A_3}+v_{A_1\\cup A_2 \\cup A_3}= v_{A_1}+v_{A_2}+v_{A_3}+v_{A_4}$ by \\cite{KM07}*{Lemma 2.6}. This yields a contribution of minus one for the weight of this cone.\n\nIn the third and fourth case, neither the codimension one cone itself nor any of its neighbors contains the vector $v_{1...j}$. Therefore it gets weight zero. \nThe claim follows.\n\\end{proof}\n\n\\begin{rem}\\label{rem:coneintersect}\nThe important consequence of this lemma is the statement that a cone with a $j+2$-valent vertex adjacent to the ends $1,\\ldots,j$ appears with weight one in the intersection $ D_{12}\\cdot D_{123}\\cdot  \\ldots \\cdot D_{1...j}$. It is straight-forward to generalize this statement to a cone $C$ with an arbitrary $j+2$-valent vertex $V$, adjacent to the edges $e_1,\\ldots,e_{j+2}$.\nWe denote by $A_i$ the subset of ends that can be reached from $V$ via $e_i$. Then $C$ appears with weight one in the intersection $ D_{A_1\\cup A_2}\\cdot D_{A_1\\cup A_2 \\cup A_3}\\cdot  \\ldots \\cdot D_{A_1\\cup \\ldots\\cup A_{j-1}}$.\nAlso, to cut out a cone  with several higher-valent vertices, we can combine several such intersection products.\n\\end{rem}\n\n\n\n\\begin{lemma}\\label{lem:pullbackintersect}\n The intersection $\\Psi_\\alpha\\cdot \\ft_\\alpha^\\ast( D_{12})\\cdot\\ft_\\alpha^\\ast( D_{123})\\cdot   \\ldots \\cdot \\ft_\\alpha^\\ast(D_{1...j})$ for some $j\\geq 2$ in $\\mathcal{M}_{0,m+1}$ consists of all cones corresponding to a type with the ends $1,\\ldots,j$ \n\\begin{itemize}\n \\item and $\\alpha$ at a $j+3$-valent vertex with weight $j$, \n\\item at a $j+2$-valent vertex, and $\\alpha$ adjacent to some other vertex with weight $1$,\n\\item and $\\alpha$ at a $j+2$-valent vertex adjacent to a $4$-valent vertex with weight $-(j-1)$,\n\\item at a $j+1$-valent vertex adjacent to a $5$-valent vertex with $\\alpha$ with weight $-2$,\n\\item at a $j+1$-valent vertex adjacent to a $4$-valent vertex and $\\alpha$ adjacent to some other vertex with weight $-1$.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n The proof is again by induction. For $j=2$, we intersect $\\Psi_\\alpha$ with $\\ft_\\alpha^\\ast(\\varphi_{12})=\\varphi_{12}\\circ \\ft_\\alpha$. This map sends the vector $v_{12}$ and the vector $v_{12\\alpha}$ to one and all other $v_i$ to zero. Codimension one cones of $\\Psi_\\alpha$ either have a $5$-valent vertex with $\\alpha$ or a $4$-valent vertex with $\\alpha$ and another $4$-valent vertex. In the first case, if $1$ and $2$ are also adjacent to the $5$-valent vertex, two of the six neighbors are spanned by vectors mapping to one, so we get weight $2$. In the second case, the vector $v_{12\\alpha}$ is contained in the codimension one cone itself and appears in the sum of the vectors spanning the neighbors if $1$ and $2$ are adjacent to the vertex with $\\alpha$ and the other $4$-valent vertex is adjacent. Such a cone then comes with weight minus one.\n\nFor the induction step, assume the statement is true for $j-1$. The codimension one cones of $\\Psi_\\alpha\\cdot \\ft_\\alpha^\\ast( D_{12})\\cdot\\ft_\\alpha^\\ast( D_{123})\\cdot   \\ldots \\cdot \\ft_\\alpha^\\ast(D_{1...j})$ then consist of all cones corresponding to a type with the ends $1,\\ldots,j-1$ \n\\begin{enumerate}\n \\item and $\\alpha$ at a $j+3$-valent vertex, \n\\item and $\\alpha$ at a $j+2$-valent vertex and one $4$-valent vertex,\n\\item at a $j+2$-valent vertex, $\\alpha$ somewhere else ,\n\\item at a $j+1$-valent vertex and a $5$-valent vertex with $\\alpha$,\n\\item at a $j+1$-valent vertex and a $4$-valent vertex, $\\alpha$ somewhere else,\n\\item and $\\alpha$ at a $j+1$-valent vertex, next to a $5$-valent vertex,\n\\item and $\\alpha$ at a $j+1$-valent vertex next to a $4$-valent vertex and another $4$-valent vertex,\n\\item at a $j$-valent vertex next to a $6$-valent vertex with $\\alpha$,\n\\item at a $j$-valent vertex next to a $5$-valent vertex with $\\alpha$ and another $4$-valent vertex,\n\\item at a $j$-valent vertex next to a $5$-valent vertex and $\\alpha$ somewhere else,\n\\item at a $j$-valent vertex next to a $4$-valent vertex, and another $4$-valent vertex and $\\alpha$ somewhere else. \n\\end{enumerate}\nFor the cases (6)-(11) it is easy to see that neither the codimension one cone itself nor any of its neighbors contains the vectors $v_{1..j}$ or $v_{1..j\\alpha}$ which are mapped to one by $\\ft_\\alpha^\\ast(\\varphi_{1..j})=\\varphi_{1..j}\\circ \\ft_\\alpha$. Thus any of these cones is taken with weight zero.\n\nIn the first case, if $j$ is also adjacent to the $j+3$-valent vertex, we have a neighbor spanned by $v_{1...j\\alpha}$ with weight $j-1$, and a neighbor spanned by $v_{1...j}$  with weight one. Altogether the weight  is $j$.\n\nIn the second case, if $j$ is also adjacent to the $j+2$-valent vertex the vector $v_{1...j\\alpha}$ is contained in the codimension one cone itself. If in addition the $4$-valent vertex is adjacent to the $j+2$-valent vertex, the vector $v_{1...j\\alpha}$ appears in the sum of the three neighbors corresponding to the resolutions of the $4$-valent vertex. Since any such resolution comes with weight $j-1$, this codimension one cone has weight $-(j-1)$.\n\nIn the third case, if $j$ is also adjacent to the $j+2$-valent vertex, we have one neighbor spanned by $v_{1...j}$  with weight one, so we also get weight one for this cone.\n\nIn the fourth case, none of the resolutions of the $j+1$-valent vertex contains the vectors $v_{1...j}$ or $v_{1...j\\alpha}$.\n The vector $v_{1...j}$ is contained in the cone itself however if $j$ is also adjacent to the $j+1$-valent vertex. We can also resolve the $5$-valent with $\\alpha$ in such a way that $\\alpha$ is still at a $4$-valent vertex. All these six neighbors have weight one. Denote the four edges not equal to the end $\\alpha$ but adjacent to the $5$-valent vertex by $e_1,\\ldots,e_4$ and denote by $A_i$ the subset of ends that can be reached from the $5$-valent vertex via $e_i$. Then the six neighbors are spanned by the vectors $v_{A_1\\cup A_2}$, $v_{A_1\\cup A_3}$, $v_{A_2\\cup A_3}$, $v_{A_1\\cup A_2\\cup\\{\\alpha\\}}$, $v_{A_1\\cup A_3\\cup\\{\\alpha\\}}$ and $v_{A_2\\cup A_3\\cup\\{\\alpha\\}}$ whose sum equals $2v_{A_1}+2v_{A_2}+2v_{A_3}+2\\cdot v_{A_1\\cup A_2\\cup A_3\\cup\\{\\alpha\\}}=2v_{A_1}+2v_{A_2}+2v_{A_3}+2\\cdot v_{A_4}$ by \\cite{KM07}{Lemma 2.6}. Thus we get weight $-2$ if and only if $A_i=\\{1,\\ldots,j\\}$ for $i=1,2,3$ or $4$ which is the case if and only if the $5$-valent vertex with $\\alpha$ is adjacent to the $j+1$-valent vertex with $1,\\ldots,j$.\n\nThe fifth case is analogous to the second case of Lemma \\ref{lem:divintersect}. All neighbors have weight one. Therefore we get weight minus one if the $4$-valent vertex is adjacent to the $j+1$-valent vertex and $j$ is adjacent to the $j+1$-valent vertex. The claim follows.\n\\end{proof}\n\n\n Now  we intersect $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ with the codimension $k$-skeleton of $\\mathcal{M}_{0,n}$. Let $K$ denote a cone of the codimension $k$-skeleton. It corresponds to a combinatorial type of a tree $\\Gamma$ with $r-k$ vertices $V_1,\\ldots,V_{r-k}$ of valence $\\val(V_i)=k_i$ with $\\sum (k_i-3)=k$. \nRemark \\ref{rem:coneintersect} tells us how to pick functions $\\varphi_1,\\ldots,\\varphi_k$ that cut out $K$ with weight one. Thus we want to compute \n$\\mathbb{H}^{\\trop}_k(\\mathbf{x})\\cdot\\varphi_1\\cdot\\ldots\\cdot\\varphi_k $ locally around $K$. We have\n\\begin{align*}\n& \\mathbb{H}^{\\trop}_k(\\mathbf{x})\\cdot\\varphi_1\\cdot\\ldots\\cdot\\varphi_k = \\\\\n& \\ft_{\\ast}\\Big(\\Psi_{n+1}\\cdot \\prod_{i=n+2}^{n+r-k} \\big(\\Psi_i \\cdot \\ev_i^{\\ast}(p_i) \\big) \\Big) \\cdot\\varphi_1\\cdot\\ldots\\cdot\\varphi_k =\\\\\n& \\ft_{\\ast}\\Big(  \\Psi_{n+1}\\cdot \\prod_{i=n+2}^{n+r-k} \\big(\\Psi_i \\cdot \\ev_i^{\\ast}(p_i) \\big) \\cdot\\ft^\\ast(\\varphi_1)\\cdot\\ldots\\cdot\\ft^\\ast(\\varphi_k)    \\Big) =\\\\\n& \\ft_{\\ast}\\Big(  \\prod_{i=n+1}^{n+r-k}\\Psi_{i} \\cdot\\ft^\\ast(\\varphi_1)\\cdot\\ldots\\cdot\\ft^\\ast(\\varphi_k)  \\cdot \\prod_{i=n+2}^{n+r-k}  \\ev_i^{\\ast}(p_i)  \\Big)\n\\end{align*}\nwhere the second equality holds by the projection formula \\cite{AR07}*{Proposition 4.8}.\n\nTo get a nonzero intersection of $\\prod_{i=n+1}^{n+r-k}\\Psi_{i} \\cdot\\ft^\\ast(\\varphi_1)\\cdot\\ldots\\cdot\\ft^\\ast(\\varphi_k) $ with the cycle $ \\prod_{i=n+2}^{n+r-k}  \\ev_i^{\\ast}(p_i) $, the ends $n+1,\\ldots, n+r-k$ must be adjacent to different vertices. The type $\\Gamma$ corresponding to $K$ has $r-k$ vertices, and so we can attach one new end to each vertex of $\\Gamma$. There are $m(\\Gamma)$ ways to do this, where we use the notation from Definition \\ref{def:weights} (we do not need to pick moving vertices here). Hence $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ intersects $K$ in $m(\\Gamma)$ points with a nonzero weight.\nEach such point is the push-forward of an intersection point of  $ \\prod_{i=n+2}^{n+r-k}  \\ev_i^{\\ast}(p_i) $ with a cone $\\tilde{K}$ of $\\prod_{i=n+1}^{n+r-k}\\Psi_{i}$ where all ends are adjacent to different vertices. To compute the weight of $\\tilde{K}$ in $\\prod_{i=n+1}^{n+r-k}\\Psi_{i} \\cdot\\ft^\\ast(\\varphi_1)\\cdot\\ldots\\cdot\\ft^\\ast(\\varphi_k) $, we use a generalization of Lemma \\ref{lem:pullbackintersect} analogous to Remark \\ref{rem:coneintersect}: in $\\tilde{K}$, we have $r-k$ vertices $V_i$ of valence $k_i+1$ each of which is adjacent to an end with a Psi-class condition. We thus get  weight $\\prod (k_i-2)$. From Lemma \\ref{lem-evproduct}, the further intersection with $ \\prod_{i=n+2}^{n+r-k}  \\ev_i^{\\ast}(p_i) $ yields a factor equal to the product of weights of all bounded edges.\nWe have thus proved the following statement that again illustrates the analogy between classical and tropical Hurwitz loci (compare with Lemma \\ref{coeff}):\n\n\\begin{prop}\n Let $K$ be a cone of $\\mathcal{M}_{0,N}$ of codimension $k$, corresponding to the type $\\Gamma$.\nThen the intersection $\\mathbb{H}^{\\trop}_k(\\mathbf{x})\\cdot K$ consists of $m(\\Gamma)$ points, each with weight $\\prod_v (\\val(v)-2)\\cdot \\varphi(\\Gamma)$\nwhere the product goes over all vertices $v$ of $\\Gamma$.\n\\end{prop}\n\n\n\n\\section{Wall Crossings}\n\n\n\nWe have seen that Hurwitz cycles are polynomials in each chamber $\\mathfrak{c}$. In this section we investigate wall-crossings, i.e. how the cycles change from chamber to chamber. \n\\begin{defn}\nLet $I\\subseteq \\{1, \\ldots, n\\}$ and consider the wall $W_I=\\{\\sum_{i\\in I}x_i =0\\} $. Let $\\mathfrak{c}^+$  and $\\mathfrak{c}^-$ be two adjacent chambers: $\\sum_{i\\in I}x_i >0$ in $\\mathfrak{c}^+$,  $\\sum_{i\\in I}x_i < 0$ in $\\mathfrak{c}^-$ and for every $J\\not=I \\subseteq  \\{1, \\ldots, n\\}$ the sign of $ \\sum_{i\\in J}x_i$ is the same in both chambers. Let $\\mathbb{H}_k^+(\\mathbf{x})$ (resp. $\\mathbb{H}_k^-(\\mathbf{x})$) be the polynomial class giving the Hurwitz cycle in $\\mathfrak{c}^+$(resp. $\\mathfrak{c}^-$). By {wall crossing formula} at the wall $I$ we mean the formal difference of cycles:\n\\begin{equation}\nWC_{I,k}(\\mathbf{x}):= \\mathbb{H}_k^+(\\mathbf{x}) -\\mathbb{H}_k^-(\\mathbf{x}) \\in Z_k(\\overline{M}_{0,n}).\n\\end{equation}\n\n\\end{defn}\nNaturally the difference of two polynomial cycles is a polynomial cycle: the upshot is that such a cycle can be expressed inductively in terms of Hurwitz cycles.\n\n\\begin{note} Let $\\mathbf{x}$ and $\\mathbf{y}$ be two tuples of integers such that $\\sum{x_i}=-\\sum{y_j}=\\epsilon\\not=0$. Consider the Hurwitz cycles $\\mathbb{H}_{k_1}(\\mathbf{x},-\\epsilon)\\in \\overline{M}_{0,n_1+1}$ and $\\mathbb{H}_{k_2}(\\mathbf{y},\\epsilon)\\in \\overline{M}_{0,n_2+1}$ and the gluing morphism \n$\\gl:\\overline{M}_{0,n_1+1}\\times\\overline{M}_{0,n_2+1}\\to \\overline{M}_{0,n_1+n_2}$. We denote:\n$$\n\\mathbb{H}_{k_1}(\\mathbf{x},-\\epsilon)\\boxtimes\\mathbb{H}_{k_2}(\\mathbf{y},\\epsilon):= \\gl_\\ast\\left( \\mathbb{H}_{k_1}(\\mathbf{x},-\\epsilon)\\times\\mathbb{H}_{k_2}(\\mathbf{y},\\epsilon)\\right) \\in Z_{k_1+k_2}( \\overline{M}_{0,n_1+n_2}).\n$$\n\\end{note}\n\nWith this notation in place we are ready to state the wall crossing formulas.\n\n\\begin{thm}[Classical Wall Crossing]\n\\label{thm:wc}\nLet $I\\subseteq \\{1, \\ldots, n\\}$ and consider the wall $W_I=\\{\\epsilon:=\\sum_{i\\in I}x_i =0\\} $. Then:\\begin{equation}\n\\label{eq:wc}\nWC_{I,k}(\\mathbf{x})= \\epsilon \\sum_{j=\\max\\{0,1+k-r_2\\}}^{\\min\\{k,r_1-1\\}}  {{r-k}\\choose{|I|-1-j}}\\mathbb{H}_j(\\mathbf{x}_I,-\\epsilon)\\boxtimes\\mathbb{H}_{k-j}(\\mathbf{x}_{I^c},\\epsilon)\n\\end{equation}\n\\end{thm}\n\\begin{proof}\nThe proof of Theorem \\ref{thm:wc} is parallel to \\cite{CJM10}*{Theorem 6.10}.\nWe first remark that the bounds of the summation are simply  recording the fact that $j$ (resp. $k-j$) must be less than or equal that the dimension of $\\overline{M}_0^{\\sim}(\\mathbf{x}_I,-\\epsilon)$ (resp. $\\overline{M}_0^{\\sim}(\\mathbf{x}_{I^c},\\epsilon)$). One may make the summation simply from $0$ to $k$ by noting that the Hurwitz loci of dimension greater than the corresponding moduli spaces of maps are empty.\n\nHurwitz cycles are completely described by the tropical dual graphs of the boundary strata in the moduli spaces of maps. In order for a tropical dual graph to contribute to the wall crossing, it must have an edge with weight equal to the equation of the wall. For a given tropical dual graph $\\Gamma$, if such an edge exists, then it is unique and we call it the special edge. \n\nCutting the special edge separates the graph into two subtrees  $\\Gamma_I$ and $\\Gamma_{I^c}$. \nThe ends  of $\\Gamma_I$ (resp. $\\Gamma_{I^c}$) are labelled by $x_i\\in I$ and $-\\epsilon$ (resp. $x_i\\in I^c$ and $\\epsilon$). We note immediately that $\\Gamma_I,\\Gamma_{I^c}$ are a pair of graphs identifying a boundary stratum appearing in the product of  Hurwitz cycles on the right hand side of formula \\eqref{eq:wc}. We make this connection more precise in order to extract quantitative information.\n\nFor $0\\leq j\\leq k$, let\n$\nR_j=\\left\\{ \\left(\\Gamma_1, \\Gamma_2, \\mathfrak{m} \\right)\\right\\},\n$\nwhere:\n\\begin{itemize}\n\\item $\\Gamma_1$ is the tropical dual graph of a stratum in  $\\tilde{\\mathbb{H}}_j(\\mathbf{x}_I,-\\epsilon)$ (pushing forward non-trivially to $\\overline{M}_{0,n}$).\n\\item $\\Gamma_2$ is the tropical dual graph of a stratum in  $\\tilde{\\mathbb{H}}_{k-j}(\\mathbf{x}_{I^c},\\epsilon)$ (pushing forward non-trivially to $\\overline{M}_{0,n}$).\n\\item $\\mathfrak{m}$ is a total ordering of the vertices of $\\Gamma_1 \\cup \\Gamma_2$, compatible with the total ordering of the vertices of  $\\Gamma_1$ and $\\Gamma_2$.\n\\end{itemize}\n\nCutting the special edge gives a function $\\Cut$ from the set of graphs contributing to the wall crossing formula to the union $R= \\cup_{j=0}^k R_j$. We claim that $\\Cut$ is a bijection, and it will come as little  surprise that the inverse function  $\\Glue$ consists in gluing  the two graphs along the special edge labelled $\\pm \\epsilon$. The total ordering $\\mathfrak{m}$ is precisely the information needed to make such gluing well defined. We note in particular that $\\mathfrak{m}$ determines in which direction the special edge is pointing once it is glued.\n\nGiven a graph  $\\Gamma$ contributing to the wall crossing, we note that the multiplicity it contributes to the wall crossing formulas is $\\epsilon$ times the product of all weights of all non-special internal edges (this is obvious if $\\Gamma$ comes from $\\mathbb{H}_k^+(\\mathbf{x})$. If  $\\Gamma$ comes from $\\mathbb{H}_k^-(\\mathbf{x})$, then the weight of the special edge is $-\\epsilon$, and there is another minus sign coming from the wall crossing formula). On the other hand the pair of graphs $\\Gamma_1$ and $\\Gamma_2$ in $\\Cut(\\Gamma)$ have multiplicity in $\\mathbb{H}_j(\\mathbf{x}_I,-\\epsilon)\\boxtimes\\mathbb{H}_{k-j}(\\mathbf{x}_{I^c},\\epsilon)$ equal to the product of all non-special internal edges of $\\Gamma$. Therefore $\\Cut$ is a bijection that preserves the multiplicities on both sides of formula \\eqref{eq:wc}.\n\n The proof is then concluded by remarking that if $\\Gamma_1, \\Gamma_2$ appear in $R_j$, there are ${{r-k}\\choose{|I|-1-j}}$ possible ways of giving a total ordering of the vertices of $\\Gamma_1 \\cup \\Gamma_2$, compatible with the total ordering of the vertices of  $\\Gamma_1$ and $\\Gamma_2$. \n\\end{proof}\n\nThe wall crossing formula on the tropical side is similar: the only apparent difference is the lack of the multiplicative factor $\\epsilon$, reflecting the fact that polynomiality does not appear in the generic representative of a tropical Hurwitz cycle.\nDifferently from the classical side however, it is not only the weights that depend on $\\mathbf{x}$ but the cycles themselves, making even the statement of a wall crossing formula more subtle.\n\nFix a wall $W_I$ and two adjacent chambers $\\mathfrak{c}^+$  and $\\mathfrak{c}^-$ with $\\epsilon:=\\sum_{i\\in I}x_i >0$ in $\\mathfrak{c}^+$,  $\\epsilon < 0$ in $\\mathfrak{c}^-$. We denote by $\\mathbb{H}^{\\trop,+}_k(\\mathbf{x})$ resp.\\ $\\mathbb{H}_k^{\\trop,-}(\\mathbf{x})$ the Hurwitz cycles in the two chambers.\nWe would like to evaluate  both $\\mathbb{H}^{\\trop,+}_k(\\mathbf{x})$  and $\\mathbb{H}_k^{\\trop,-}(\\mathbf{x})$\n at $\\mathbf{x}\\in \\mathfrak{c}^+$\n\n and then consider the difference. However the fact that $\\epsilon$ changes sign when crossing the wall requires some interpretation, since the edge lengths of tropical covers\nare required to be positive. \nConsider a point in $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ corresponding to a tropical cover with an edge of weight $-\\epsilon$ connecting two vertices that are mapped to two points $p<q$ in $\\TP$. The length of the special edge of this tropical cover then equals $\\frac{q-p}{-\\epsilon}$, which is positive in $\\mathfrak{c}^-$, but negative in $\\mathfrak{c}^+$. In the latter case, $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ does not live in  $\\mathcal{M}_{0,n}$ but in the real vector space $\\mathbb{R}\\mathcal{M}_{0,n}$ surrounding it.\nTo define $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})\\subset \\mathcal{M}_{0,n}$ for $\\mathbf{x}\\in \\mathfrak{c}^+$, we consider  the linear map $c_\\epsilon$ that maps $v_I$ to itself for all $I\\neq \\epsilon$ and $v_\\epsilon$ to $-v_\\epsilon$, and redefine $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ to be the image of $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})\\subset \\mathbb{R}\\mathcal{M}_{0,n}$  under $c_\\epsilon$. \n\n\nAlso the weights of the cycle $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ require some interpretation: in Lemma \\ref{lem:cellsweights} we have seen that the weight of a cell equals the gcd of products of weights of trees where we shrink edges in such a way that all moving vertices merge with a non-moving vertex. When computing $\\mathbb{H}_k^{\\trop,-}(\\mathbf{x})$, we use the convention to use the negative gcd whenever $-\\epsilon$ appears in a product.\n\n\\begin{defn}\nWith the notation from the above paragraph, we now define the \\emph{tropical Wall Crossing} as\n$$WC^{\\trop}_{I,k}(\\mathbf{x}):= \\mathbb{H}^{\\trop,+}_k(\\mathbf{x}) - \\mathbb{H}^{\\trop,-}_k(\\mathbf{x}).$$\n\\end{defn}\n\n\n\\begin{example}\\label{ex:wcm05}\n In Example \\ref{ex:m05hurwitzcurve}, we  consider the Hurwitz curve for  $\\mathbf{x}=(x_1,\n\\ldots,x_5)$ in $\\mathfrak{c}^+$ defined by $x_1, x_2>0$, $x_3,x_4,x_5<0$, $x_1>|x_i+x_j|$ for  $i,j \\in\\{3,4,5\\}$ and $x_2<-x_i$ for  $i=3,4,5$ in $\\mathcal{M}_{0,5}$. We now cross the wall $\\epsilon:=x_1+x_4+x_5=0$. \n\n$\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) $ and $ \\mathbb{H}^{\\trop,-}_1(\\mathbf{x})$ only differ in cones whose corresponding type contains an edge with weight $\\pm\\epsilon$, i.e.\\ in any cone containing the vector $v_{23}$. There are three such cones. For two of these cones, the Hurwitz curve $\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) $ looks as depicted in Figure \\ref{fig:m052} and for the third cone as in Figure \\ref{fig:m054}. In $ \\mathbb{H}^{\\trop,-}_1(\\mathbf{x})$, any edge with weight $-\\epsilon$ changes direction. In the cone spanned by $v_{23}$ and $v_{14}$, if we choose the vertex adjacent to end $5$ to be the moving vertex, we now have two ways to order the remaining vertices, both corresponding to linear ends. Thus one linear edge is replaced by two linear ends in this cone. The same is true for the cone spanned by $v_{23}$ and $v_{15}$ by symmetry. In the cone spanned by $v_{23}$ and $v_{45}$ contrarily, if the moving vertex is adjacent to end $1$, we have only one way to order the remaining vertices corresponding to a linear edge instead of the two linear ends (see Figures \\ref{fig:m053} and \\ref{fig:m054}). \nNote also that the constant ends of direction $v_{23}$ do not have a factor of $\\epsilon$ in their weight, thus they appear with the same sign both in $\\mathbb{H}^{\\trop,+}(_1\\mathbf{x}) $ and $ \\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ and cancel in the difference. The other constant ends have weight $\\epsilon$ in $\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) $ but weight $-\\epsilon$ in $ \\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$, so they do not cancel but add up to a constant end with weight $2\\epsilon$. Figure \\ref{fig:wc} depicts the tropical wall crossing curve for these two chambers. Blue edges are edges of $\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) $, red edges are edges of $ \\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$. The green constant ends appear in both and add up to the weight $2\\epsilon$. The picture only shows the three cones of $\\mathcal{M}_{0,5}$ in which the difference $\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) - \\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ is nonzero.\n\n\n\\begin{figure}[tb]\n\\input{wc.pstex_t}\n\\caption{The wall crossing curve in $\\mathcal{M}_{0,5}$ for $\\mathfrak{c}^+$ and $\\mathfrak{c}^-$.}\n\\label{fig:wc}\n\\end{figure}\n\n\\end{example}\n\n\nAs in the classical world, we want to describe a tropical wall crossing in terms of cutting and regluing. \n Any cell contributing to the wall crossing parameterizes graphs with a special edge that we may cut to obtain two subgraphs each of which is a tropical cover of the projective line.  Remembering the length of the edge that gets cut accounts for the fact that we want to mod out by translations only once. We make this process precise in the following paragraph.\n\n\n\\begin{construction}\n \nLet $I\\subseteq \\{1, \\ldots, n\\}$ and consider the wall $W_I=\\{\\epsilon:=\\sum_{i\\in I}x_i =0\\} $. \nAssume that $|I|=n_1$ and $|I^c|=n_2$ and denote $r_i=n_i-1$ for $i=1,2$.\nLet $\\Gamma$ be a tropical cover in a $k$-dimensional cell that contributes to the wall crossing, and hence it contains an edge $e$ with weight $\\pm\\epsilon$. Cutting $e$ we obtain two subgraphs $\\Gamma_1$ and $\\Gamma_2$ that are themselves tropical covers of the projective line. We assume that $\\Gamma_1$ contains the ends in $I$ and $\\Gamma_2$ contains the ends in $I^c$. Both $\\Gamma_1$ and $\\Gamma_2$ have an extra end that we denote by $E_1$ (resp.\\ $E_2$). According to our conventions ends are oriented inward, and hence the balancing condition gives $E_1$ weight $-\\epsilon$ and $E_2$ weight $\\epsilon$. \nAssuming that $r_1-j$ fixed vertices are in $\\Gamma_1$,  we can interpret $\\Gamma_1$ as an element in $\\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))$ and $\\Gamma_2$ as an element in $\\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon))$ (adjusting the labeling of the ends). We wish to remember the length of the edge we cut and whether $\\Gamma$ belonged to $\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) $ or $\\mathbb{H}^{\\trop,-}_1(\\mathbf{x}) $, so we define:\n$\n\\Cut (\\Gamma) \\in \\left(\\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))\\times {\\mathbb R}\\right)\\times \\left(\\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon))\\times {\\mathbb R}\\right)\n$\nby \n\n\\begin{equation} \n\\label{eq:cut}\n\\Cut(\\Gamma) =\\begin{cases} \\left((\\Gamma_1, 0) , (\\Gamma_2, l(e))\\right) & \\Gamma \\in  \\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) \\\\  \\left((\\Gamma_1, 0) , (\\Gamma_2, -l(e))\\right) & \\Gamma \\in  \\mathbb{H}^{\\trop,-}_1(\\mathbf{x}). \\end{cases}\n\\end{equation}\n\n\n\n\n\\begin{rem} \nOur choice to have two ${\\mathbb R}$ coordinates and assigning one of them to be $0$ seems, and in fact is, somewhat arbitrary. However, it will be handy when comparing weights of the same cells appearing on opposite sides of the wall crossing formula.\n\\end{rem}\n\\begin{rem}\nIn order for  equation \\eqref{eq:cut} to make sense we need to drop Convention \\ref{conv} (see page \\pageref{conv}) and remember tropical covers are equivalent up to translation. We therefore use one of the vertices in each of the subgraphs to fix a parameterization of $\\TP$.\n\\end{rem}\n\nWe reverse this operation to glue two graphs in  $\\left(\\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))\\times {\\mathbb R}\\right) \\times\\left(\\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon))\\times {\\mathbb R}\\right)$. Denote by $l_i$ the ${\\mathbb R}$ coordinate function for the $i$-th factor in the product.\nIf $V_i$ is the interior vertex of $\\Gamma_i$ adjacent to $E_i$, then  define $\\ev_{E_i}: \\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))\\times {\\mathbb R}\\to {\\mathbb R}$ by $$\\ev_{E_i}(\\Gamma_i, l_i)=\\ev_{V_i}+(-1)^il_i\\cdot \\epsilon. $$\n\n\n\nWe can glue any two pieces in $(\\ev_{E_1}-\\ev_{E_2})^\\ast(0)\\cdot (\\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))\\times {\\mathbb R}) \\times (\\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon))\\times {\\mathbb R}) $ to one tropical cover in $\\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x})$. To make this operation the inverse to $\\Cut$ defined above, we further impose $l_1=0$. \n\n\nThe above discussion shows that \n\\begin{align*}\\mathcal{G} := (\\ev_{E_1}-\\ev_{E_2})^\\ast(0)\\cdot l_1^\\ast(0) \\cdot  \\Big( &(\\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))\\times {\\mathbb R}) \\times \\\\  &  (\\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon))\\times {\\mathbb R}) \\Big)\\end{align*}\n is in bijection to the set of covers in $\\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x})$ contributing to the wall crossing \n(see \\cite{GMO} for a similar gluing construction for moduli spaces). \n\nWe now define the folding map $\\fold:\\mathcal{G} \\rightarrow \\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x})$ that maps $l_2$ to its absolute value.\nThe folding map is not globally a tropical morphism, it is only locally a morphism away from $\\mathcal{G}\\cdot l_2^\\ast(0)$.\nConsequently, while $\\mathcal{G}$\nis a tropical variety, its image under $\\fold$ is not. Since the wall crossing curve is also not a tropical variety however, this is not disturbing.\n\nTo make the image of $\\fold$ a weighted polyhedral complex, we give each cell the sum of the weights of its preimages (cells are subdivided by \n$\\mathcal{G}\\cdot l_2^\\ast(0))$. This coincides with the weight of the push forward of $\\fold$ locally where it is a morphism.\n\\end{construction}\n\n\n\nWe  now state the first version of the tropical wall crossing. \n \\begin{prop}[Tropical Wall Crossing, first version]\nLet $I\\subseteq \\{1, \\ldots, n\\}$ and consider the wall $W_I=\\{\\epsilon:=\\sum_{i\\in I}x_i =0\\} $. Then:\n\n\n\n\n\n\n\n\\begin{equation}\\hspace{-1cm}\n\\begin{array}{cl}\nWC^{\\trop}_{I,k}(\\mathbf{x}) &=  \\ft_{\\ast} \\bigg(  \\sum_{j=\\max\\{0,1+k-r_2\\}}^{\\min\\{k,r_1-1\\}} \\;\\;\\;\\;  \\sum_{n+1 \\leq i_1<\\ldots<i_{r_1-j}\\leq n+r-k}  \n \\fold   \\Big(  \\\\ & \\\\\n & \\left.\\left.  \\hspace{-1cm}\n\\Big( \\Psi_{n_1+2}\\cdot \\prod_{l=2}^{r_1-j} \\big(\\Psi_{l+n_1+1} \\cdot \\ev_{l+n_1+1}^{\\ast}(p_{i_{l}}) \\big)      \n\\cdot \\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon)) \\Big) \\times {\\mathbb R}  \n\\times   \\right.\\right.\\\\ & \\\\\n& \\left.\\left.\\hspace{-1cm}\n\\Big( \\Psi_{n_2+2}\\cdot \\prod_{l=2}^{r_2-(k-j)} \\big(\\Psi_{l+n_2+1} \\cdot \\ev_{l+n_2+1}^{\\ast}(p_{j_{l}}) \\big)   \\cdot \\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon)) \\Big) \\times {\\mathbb R} \\times \\right.\\right.\\\\ & \\\\\n&\\hspace{-1cm}\n (\\ev_{E_1}-\\ev_{E_2})^\\ast(0)\\cdot l_1^\\ast(0)\n\\Big)\\bigg), \n\\end{array}\n\\label{twc-one}\n\\end{equation}\nwhere $j_1<\\ldots<j_{r_2-(k-j)}$ are defined by $$\\{n+1,\\ldots,n+r-k\\}=\\{i_1,\\ldots,i_{r_1-j}, j_1,\\ldots, j_{r_2-(k-j)}\\}.$$ \n\n\\end{prop}\n\n\\begin{proof}\n We first show that graphs where the special edge separates two pieces $\\Gamma_1$ and $\\Gamma_2$ such that there is no fixed (i.e. $4$-valent) vertex in $\\Gamma_1$ do not contribute to the wall crossing. \n Analogously the same is true for types with no $4$-valent vertex in $\\Gamma_2$. \n This justifies the bounds of the first sum: $j \\geq 1+k-r_2$ is equivalent to $r_2-(k-j)\\geq 1$ which means that there is at least one $4$-valent vertex in $\\Gamma_2$, the upper bound $j\\leq r_1-1$ guarantees that there is at least one $4$-valent vertex in $\\Gamma_1$. \n\nAssume $\\Gamma$ is a type with no $4$-valent vertex in $\\Gamma_1$. Then there are only moving vertices in $\\Gamma_1$. The type $\\Gamma$ appears both in $\\mathbb{H}_k^{\\trop,+}(\\mathbf{x}) $ and in $\\mathbb{H}_k^{\\trop,-}(\\mathbf{x}) $ : the special edge can point in any direction if none of the vertices past  the special edge  are free to move. The corresponding cells are identical set-theoretically.\n To compute their weight, we have to consider types $\\Gamma'$ where all moving vertices merge with some fixed vertices. In any such type, the special edge must be shrunk since there are only moving vertices on one side of it. Hence the weight which equals the gcd of products of weights of possible $\\Gamma'$ is identical for both cells, which thus cancel in the difference $\\mathbb{H}_k^{\\trop,+}(\\mathbf{x}) - \\mathbb{H}_k^{\\trop,-}(\\mathbf{x})$.\n\nThe above construction shows that  formula \\eqref{twc-one} holds set-theoretically. \nIt remains to show that also the weights agree. Since the cells of $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ are in weight-preserving bijection with the cells of the push forward $\\mathbb{H}^{\\trop}_k(\\mathbf{x}) \\subset \\mathcal{M}_{0,n}$, we can show the equality of the weights for the cells before pushing forward with the forgetful map. Also, we can separate the contributions from the two sides of the wall, i.e. ``unfold'' on the cut-and-glue side of the equation.\nLet $K$ be a cell \ncontributing to the wall crossing\n\n  corresponding to the combinatorial type $\\Gamma$.\n \n  Following the construction above we produce $\\Gamma_1$ and $\\Gamma_2$ which  identify a unique cell $K'$ on the right hand side of equation \\eqref{twc-one}.  By Remark \\ref{rem-intersect}  the weight of $K$ is the gcd of the absolute values of the maximal minors of the matrix of the evaluation maps evaluating the fixed vertices of $\\Gamma$ in local coordinates. Analogously, the weight of the corresponding cut-and-glue cell $K'$ is the gcd of the absolute values of the maximal minors of the matrix of the evaluation maps evaluating the fixed vertices of $\\Gamma_1$, the fixed vertices of $\\Gamma_2$, of the map $\\ev_{E_1}-\\ev_{E_2}$\n  and of the map $l_1$.\nLocal coordinates of the cell $K$ are given by the lengths of all bounded edges of $\\Gamma$,\nwhereas for the cell $K'$ they are given by the bounded edges of $\\Gamma$ except the special edge, plus the lengths $l_1$ and $l_2$. The map $l_1$ produces a row for the matrix with a one in the $l_1$-column and only zeros in the other columns. Any nonzero maximal minor must contain this column and equals the determinant of the submatrix where we erase the $l_1$-column and the row for the map $l_1$. Therefore we can erase this column and row without changing any minor in the matrix for $K'$. We show that the matrix for $K$ and the reduced matrix for $K'$ coincide up to row operations. Since row operations do not change the absolute value of any maximal minor, the claim follows. In the matrix for $K$, columns are the lengths in $\\Gamma_1$, the length $l$ of the special edge and the lengths in $\\Gamma_2$. As rows we have the evaluations of the contracted ends $n+2,\\ldots,n+r-k$ which for each end $i$ are given by following the unique path in $\\Gamma$ from the reference end $n+1$ (that we require to map to $0$) to $i$.\nIn the reduced matrix for $K'$, we have the analogous columns. Assume without restriction that $i_1=n+1$, i.e.\\ the end mapping to zero is contained in $\\Gamma_1$. We evaluate all other contracted ends of $\\Gamma_1$ with respect to this reference end, i.e.\\ we follow the path from this end to any other in $\\Gamma_1$. This gives exactly the same rows for the evaluation of ends of $\\Gamma_1$. In the matrix of $K$, we have a row evaluating the end that becomes the reference end in $\\Gamma_2$ (that we fix to be at the point $p_{j_1}$). This row is exactly equal to the row evaluating $\\ev_{E_1}-\\ev_{E_2}$ --- the unique path in $\\Gamma$ connecting the two reference ends. The remaining rows evaluate ends in $\\Gamma_2$. For an end $i$ in $\\Gamma_2$, we evaluate it in $K$ by following the path from the reference end in $\\Gamma_1$ to $i$. We evaluate it in $K'$ by following the path from the reference end in $\\Gamma_2$ to $i$. These two rows just differ by the path connecting the two reference ends which by the above is also a row of both matrices. Hence we can perform row operations\n to produce the matrix for $K$ from the matrix for $K'$.\n\\end{proof}\n\n\\begin{example}\nWe establish the wall crossing formula \\eqref{twc-one} for the wall crossing curve in $\\mathcal{M}_{0,5}$ from Example \\ref{ex:wcm05}. \nThe wall $\\epsilon=x_1+x_4+x_5=0$ separates any cover $\\Gamma$ containing an edge $\\epsilon$ into $\\Gamma_1$ and $\\Gamma_2$, where $\\Gamma_1$ has the ends $x_1,x_4,x_5$ and $-\\epsilon$ and $\\Gamma_2$ has the ends $x_2$, $x_3$ and $\\epsilon$. We have two contracted marked ends. Since both pieces must contain at least one to have at least one $4$-valent vertex, both contain exactly one. \nWe first assume that the end $6$  (required to map to zero) is in $\\Gamma_1$. For $\\Gamma_2$, there is exactly one type, where all four ends (including the contracted end $7$) are incident to one $4$-valent vertex. We have  a copy of ${\\mathbb R}$ parametrizing all possibilities with coordinate $l_2$.\n\nFor $\\Gamma_1$, there are six possible types, since there are three types in $\\mathcal{M}_{0,4}$ each of which has two $3$-valent vertices, and we can decide which of the two vertices should be equipped with the contracted end $6$ and become $4$-valent. \nWe neglect the length of the extra end $l_1$ that is required to be zero.\nWe  have a ray for each of the six types, the local coordinate is the length of the bounded edge that we denote by $l$. \nFigure \\ref{fig:6types} shows the six types together with the part to which they are glued.\n\n\\begin{figure}[tb]\n\\input{6types.pstex_t}\n\\caption{The possible types for $\\Gamma_1$ and $\\Gamma_2$ after cutting, assuming $6$ is in $\\Gamma_1$.}\n\\label{fig:6types}\n\\end{figure}\n\n\n\nFor each type, we  write down the gluing equation $\\ev_{E_1}-\\ev_{E_2}=0$ in local coordinates. Remember we require $6$ to be mapped to zero and $7$ to one.\nThe equations are:\n\n\\begin{align*}\n \\mathrm{I}:\\qquad &  0+l (x_1+x_4)+\\epsilon l_2=1, \\\\\n \\mathrm{II, IV,V}:\\qquad &  0+l_2\\epsilon=1,\\\\\n \\mathrm{III}:\\qquad &0+l (x_1+x_5)+\\epsilon l_2=1, \\\\\n \\mathrm{VI}:\\qquad & 0-l (-x_4-x_5)+l_2\\epsilon=1.\n\\end{align*}\n\n\n\nThese gluing equations cut out the tropical variety depicted in Figure \\ref{fig:gluing} from $\\Psi_6\\cdot \\mathcal{M}_{0,1}(\\TP,x_1,x_4,x_5,-\\epsilon)\\times \\Psi_7\\cdot \\mathcal{M}_{0,1}(\\TP,x_2,x_3,\\epsilon)\\times {\\mathbb R}$ which consists of six rays \nWe also draw the image under the folding map. The rays that are mapped from the orthant where $l_2$ is negative via the folding map are drawn with dotted lines.\n\n\\begin{figure}[tb]\n\\input{gluing.pstex_t}\n\\caption{The variety obtained from gluing and its image under folding for the first summand.}\n\\label{fig:gluing}\n\\end{figure}\n\n\n\nAnalogously, if end $7$ is in $\\Gamma_1$ and $6$ is in $\\Gamma_2$, we have the types depicted in Figure \\ref{fig:6types} but with $6$ and $7$ exchanged.\nThe gluing equations in local coordinates are as above, but with the role of $0$ and $1$ exchanged.\nFigure \\ref{fig:gluing2} shows the tropical variety cut out by the gluing equation from $\\Psi_7\\cdot \\mathcal{M}_{0,1}(\\TP,x_1,x_4,x_5,-\\epsilon)\\times \\Psi_6\\cdot \\mathcal{M}_{0,1}(\\TP,x_2,x_3,\\epsilon)\\times {\\mathbb R}$ and its image under folding. As before, rays that appear in the positive orthant after folding are drawn as dotted lines.\n\n\\begin{figure}[tb]\n\\input{gluing2.pstex_t}\n\\caption{The variety obtained from gluing and its image under folding for the second summand.}\n\\label{fig:gluing2}\n\\end{figure}\n\n\nUnder the forgetful map, the four orthants labeled I and II go to the cone spanned by $v_{23}$ and $v_{14}$, the four orthants labeled III and IV go to the cone spanned by $v_{23}$ and $v_{15}$ and the remaining four to the cone spanned by $v_{23}$ and $v_{45}$. We get the picture  seen in  Example \\ref{ex:wcm05}.\n\n\n\\end{example}\n\n\n\n\nWe now group summands in the cut-and-glue side of the equation to give a nicer interpretation of formula \\eqref{twc-one}.\nIf we denote by $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ the preimage of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ under the forgetful map, then as in Remark \\ref{rem:equivalent} the rational equivalence class of the loci \n\\begin{align*}& \\Big( \\Psi_{n_1+2}\\cdot \\prod_{l=2}^{r_1-j} \\big(\\Psi_{l+n_1+1} \\cdot \\ev_{l+n_1+1}^{\\ast}(p_{i_{l}}) \\big)      \n\\cdot \\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon)) \\Big) \\times {\\mathbb R}  \\\\&\\sim \\tilde{\\mathbb{H}}^{\\trop}_j(\\mathbf{x}_I,-\\epsilon) \\times {\\mathbb R}\\end{align*}\n does not depend on the exact choice of $i_1<\\ldots<i_{r_1-j}$, and analogously for the other loci in the sum. Up to rational equivalence we can thus group all summands of the second sum into one just counting how many such summands there are: $\\binom{r-k}{r_1-j}=\\binom{r-k}{|I|-1-j}$. Then \\eqref{twc-one} becomes\n\\begin{align*}\n &WC^{\\trop}_{I,k}(\\mathbf{x})\\sim\\\\ & \\ft_{\\ast} \\Bigg(  \\sum_{j=\\max\\{0,1+k-r_2\\}}^{\\min\\{k,r_1-1\\}}  \n\\binom{r-k}{r_1-j}  \\fold   \\bigg( \\Big(\\tilde{\\mathbb{H}}^{\\trop}_j(\\mathbf{x}_I,-\\epsilon) \\times {\\mathbb R} \\times \\tilde{\\mathbb{H}}^{\\trop}_{k-j}(\\mathbf{x}_{I^c},\\epsilon)\\times {\\mathbb R} \\Big)\n\\\\&\n\\times \n (\\ev_{E_1}-\\ev_{E_2})^\\ast(0)\\cdot l_1^\\ast(0)\\bigg)\\Bigg).\n\\end{align*}\n\nIf we furthermore denote\n\\begin{align*}\n &\\mathbb{H}^{\\trop}_j(\\mathbf{x}_I,-\\epsilon)\\boxtimes\\mathbb{H}^{\\trop}_{k-j}(\\mathbf{x}_{I^c},\\epsilon) :=\\\\&\n\\fold   \\left( \\Big(\\tilde{\\mathbb{H}}^{\\trop}_j(\\mathbf{x}_I,-\\epsilon) \\times {\\mathbb R} \\times \\tilde{\\mathbb{H}}^{\\trop}_{k-j}(\\mathbf{x}_{I^c},\\epsilon)\\times {\\mathbb R} \\Big) (\\ev_{E_1}-\\ev_{E_2})^\\ast(0)\\cdot l_1^\\ast(0)\\right)\n\\end{align*}\nwe obtain a formula for the tropical wall crossing that looks very similar to the classical wall crossing formula  \\eqref{eq:wc}.\n\n\n\\begin{cor}[Tropical Wall Crossing, reloaded]\n\\label{cor:twc}\nLet $I\\subseteq \\{1, \\ldots, n\\}$ and consider the wall $W_I=\\{\\epsilon:=\\sum_{i\\in I}x_i =0\\} $. Then:\n\\begin{align*}\n &WC^{\\trop}_{I,k}(\\mathbf{x})\\sim\\\\ & \\ft_{\\ast} \\left( \\sum_{j=\\max\\{0,1+k-r_2\\}}^{\\min\\{k,r_1-1\\}}  \\binom{r-k}{|I|-1-j}          \\mathbb{H}^{\\trop}_j(\\mathbf{x}_I,-\\epsilon)\\boxtimes\\mathbb{H}^{\\trop}_{k-j}(\\mathbf{x}_{I^c},\\epsilon) \\right).\n\\end{align*}\n\n\\end{cor}\n\n\n\n\n\n\\subsection{Hurwitz Curves: Take three}\n\nA few features of Example \\ref{ex:wcm05} can be generalized to the general one-dimensional wall crossing  case immediatly. Constant ends of direction $v_I$ cancel in the wall crossing. Constant ends of other directions can appear on both sides of the wall but will show up in the wall crossing with a positive sign. Let $\\Gamma^+$ be a graph in $\\tilde{\\mathbb{H}}^{\\trop,+}_k(\\mathbf{x})$ with a special edge and $V$ a choice of a moving vertex corresponding to a linear end. Then we have $m_V(\\Gamma^+)$ linear ends in $\\tilde{\\mathbb{H}}^{\\trop,+}_k(\\mathbf{x})$, and $m_V(\\Gamma^-)$ linear edges in $\\tilde{\\mathbb{H}}^{\\trop,-}_k(\\mathbf{x})$ (where $\\Gamma^-$ denotes the graph with the special edge reversed). All such edges come with the same weight in the wall crossing.  Analogously, $m_V(\\Gamma^+)$ quadratic edges are replaced by $m_V(\\Gamma^-)$ quadratic edges when crossing the wall, also coming with the same weight.\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWhen two liquid drops meet, a dramatic topological transformation takes place. \nA microscopic liquid neck forms between the drops, which rapidly expands due to surface tension forces (see Fig.\\ \\ref{sequenceWater}). \nThis coalescence is ubiquitous in nature on a vast range of scales, from raindrops merging in clouds \\cite{Pruppacher2010} to the coalescence of gas clouds during star formation \\cite{Montmerle2006}. \nIn industry, coalescence occurs in a multitude of situations, including viscous sintering \\cite{Djohari2009}, oil desalting \\cite{Eow2002}, dense spray systems \\cite{Ashgriz1990}, and the processing and shelf-life of emulsions \\cite{Evans1994}. \n\n\\begin{figure}[b]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.4in]{sequenceWater.pdf} \n\\end{center}\n\\caption{\nInitial moments of coalescence for two water drops of radius $A=2$ mm in air. \nFrames are 120 $\\mu$s apart. The central white spot is from the light source located behind the drops. \nA connecting neck of radius $r_{\\text{min}}$ grows from infinitesimal size until it reaches the macroscopic size of the drops. \n}\n\\label{sequenceWater}\n\\end{figure}\n\nThe singularity inherent in drop coalescence has received significant attention. \nIn the idealized limit where the drops form their initial contact over an infinitesimal region, the interfacial curvature, and therefore the pressure, diverge at the moment of contact, $t_0$. \nAlthough in reality this singularity is cut off (at least by the finite size of the molecules), it nonetheless controls the dynamics of the liquid neck over many orders of magnitude. \nThese dynamics have recently been investigated by theoretical work \\cite{Hopper1984,Hopper1990,Hopper1993a,Hopper1993b,Eggers1999,Eggers2003,Thompson2012_2}, numerical simulations \\cite{Herrera1995,MenchacaRocha2001,Eggers2003,Lee2006,Thompson2012_2,Sprittles2012}, high-speed imaging experiments \\cite{MenchacaRocha2001,Wu2004,Yao2005,Thoroddsen2005,Bonn2005,Burton2007,Yokota2011}, ultrafast x-ray phase-contrast imaging \\cite{Fezzaa2008}, and an ultrafast electrical method \\cite{Case2008,Case2009}. \nThese studies sought to understand the dynamics governing the growth of the neck radius, $r_{\\text{min}}$, as a function of time. \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.2in]{phaseDiagram.pdf}\n\\end{center}\n\\caption{\n(Color online) \nPrevious and revised phase diagram for drop coalescence in vacuum or air. \nAxes: non-dimensional neck radius, $r_{\\text{min}}\/A$, and $Oh=\\mu\/\\sqrt{\\rho \\gamma A}$ (non-dimensional viscosity). \n(a) Old picture: there are two regimes, one dominated by viscous forces where inertia can be completely ignored (Stokes regime) and one dominated by inertia. \nSolid line: $r_{\\text{min}}\/A=Oh^2$. \n(b) Revised understanding: the ``inertially limited viscous\" (ILV) regime intervenes at early times for finite-viscosity drops. \nThe Stokes and inertial regimes do not share a phase boundary. \nDashed line: Eq.\\ (\\ref{phase_boundary_3D}) with a proportionality constant of $1.4$. \nSolid line: Eq.\\ (\\ref{rcformula}). \n}\n\\label{phaseDiagram}\n\\end{figure}\n\nThe consensus from this work had been that there are just two regimes when the drops of radius $A$ are brought together in vacuum or air: (i) a highly viscous one dominated by macroscopic flows pulling the two drops together, and (ii) an inertial one described by local deformations near the growing neck. \nAt early times, drops obey purely viscous behavior, and low-viscosity drops transition into the inertial regime later on. \nThus, a coalescence phase diagram had been constructed \\cite{Eggers1999}, which is shown in Fig.\\ \\ref{phaseDiagram}(a) in terms of the dimensionless neck radius, $r_{\\text{min}}\/A$, and the Ohnesorge number, $Oh=\\mu\/\\sqrt{\\rho \\gamma A}$, which is a dimensionless viscosity (where $\\mu$ is the dynamic viscosity, $\\rho$ is the liquid density, and $\\gamma$ is the interfacial tension). \n\nRecently, Paulsen \\textit{et al.}\\ \\cite{Paulsen2012} demonstrated that a third regime---one that intervenes at early times for drops of any finite viscosity---had been missed. \nUsing high-speed imaging experiments, an ultrafast electrical method, and high-accuracy computation, they showed that at small neck radius, the surface tension force pulling the drops together is too weak to overcome the inertia of the drops. \nTherefore, coalescence \\textit{cannot} be in the Stokes-flow limit at early times, and an ``inertially limited viscous\" regime occurs. \nIn this regime, surface tension, viscous forces, and inertia all balance. \nAt late times, viscous drops ($Oh>1$) \\textit{can} reach the Stokes-flow limit, so there is an inertially limited viscous to Stokes crossover that is traversed as the neck radius grows. \n\nThe final piece of the coalescence phase diagram is the viscous-to-inertial crossover time, $\\tau_c$ (or the crossover radius, $r_c$), where the dynamics switch from the inertially limited viscous regime to a regime where only inertia is important. \nFor many fluid flows, this crossover is easily identified by computing the dimensionless Reynolds number, $Re=\\rho U L\/\\mu$, where $U$ and $L$ are characteristic velocity- and length-scales in the flows, respectively. \nCrossover behavior is expected when $Re\\approx 1$. \nFor coalescence, it was always assumed that $L=r_{\\text{min}}$ \\cite{Eggers1999,Eggers2003,Wu2004,Yao2005,Thoroddsen2005,Bonn2005,Burton2007,Case2008,Case2009,Thompson2012_2}. \n\nTo observe the viscous-to-inertial crossover, Paulsen \\textit{et al.}\\ \\cite{Paulsen2011} used an ultrafast electrical method (following refs.\\ \\cite{Case2008,Case2009}), which measures the neck radius down to tens of nanoseconds after the drops touch. \nFor salt-water drops, Paulsen \\textit{et al.}\\ observed viscous behavior more than $3$ decades later than the prediction using the accepted Reynolds number for coalescence. \nTo explain this discrepancy, they proposed that the dominant length-scale for the flows is instead given by the neck height, $L=r_{\\text{min}}^2\/A$. \nThus, a revised phase diagram for coalescence was constructed, which is pictured in Fig.\\ \\ref{phaseDiagram}(b). \n\n\nThis paper provides a more detailed experimental description and presents additional evidence for the picture developed in refs.\\ \\cite{Paulsen2012, Paulsen2011}. \nFirst, section \\ref{ExperDesc} describes the electrical method, the fluids used, and the high-speed imaging technique, and section \\ref{StokesInertialTheory} outlines several theoretical predictions for the purely viscous (Stokes) regime and the inertial regime. \n\nSection \\ref{ILVregime} provides measurements and analysis of coalescence in the Stokes regime and the inertially limited viscous regime. \nWhereas Paulsen \\textit{et al.}\\ \\cite{Paulsen2012} identified the inertially limited viscous to Stokes crossover by the motion of the back of the drops, I show that the same motions occur in the center-of-mass of the drops. \nThe neck shapes in the inertially limited viscous and Stokes regimes are consistent with two distinct similarity solutions, and the interfacial curvature at the neck minimum can be used to distinguish between the regimes. \nThe phase diagram is robust to different boundary conditions.\n\nSection \\ref{VIcrossover} provides measurements and analysis of the viscous-to-inertial crossover. \nI collapse the electrical data with a different analysis from ref.\\ \\cite{Paulsen2011}, to demonstrate that the results are not sensitive to the details of the collapse protocol. \nI argue for a new Reynolds number for coalescence, as was done in ref.\\ \\cite{Paulsen2011}, now coming from the viscous side of the transition. \nI present high-speed imaging data where the surface tension is varied, which follows the crossover scaling calculated with the new Reynolds number. \n\nSection \\ref{approach} studies the drops during their approach.\nUsing optical, electrical resistance, and capacitance measurements, I show that at low approach-speed, the drops coalesce as undeformed spheres at finite separation. \nThe data suggest that at low voltage, Van der Waals forces form the initial liquid neck (instead of forces due to the applied voltage).\nThe measurements provide an upper bound on the initial neck radius, $r_0$, which is smaller than previous estimates \\cite{Thoroddsen2005,Fezzaa2008}.\n\n\nAppendix \\ref{elecSystematic} reports checks on the electrical method, which show that the applied voltages and resulting electric fields do not affect the coalescence dynamics. Appendix \\ref{prevCross} addresses previous measurements of the viscous-to-inertial crossover in the literature. \n\nThis work gives a consistent picture wherein the inertially limited viscous regime is the asymptotic regime of liquid drop coalescence in vacuum or air. \nViscous drops ($Oh>1$) transition into the Stokes regime later on, and low-viscosity drops ($Oh<1$) crossover into the inertial regime. \nIn the inertially limited viscous regime and the inertial regime, the dominant flow gradients are on the scale of the neck height, $r_{\\text{min}}^2\/A$. \n\n\n\n\n\n\\section{Experimental Description}\n\\label{ExperDesc}\n\n\\subsection{Ultrafast electrical method}\n\nIn the experiment, two drops are formed on vertically aligned teflon nozzles of radius $A=2$ mm, which are separated by a distance $2A$. \nThe pendant drop is fixed while the sessile drop is slowly grown with a syringe pump until the drops coalesce. \nExcept for section \\ref{approach}, the experiments are at sufficiently low approach-speed ($U_{\\text{app}}< 9 \\times 10^{-5}$ m\/s) where the drops do not deform before contact.\n\n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.0in]{Schematic_Signals.pdf}\n\\end{center}\n\\caption{\n(Color online) \nElectrical method. \n(a) Coalescence cell and measurement circuit. \nLiquid hemispheres are formed on nozzles. \nOne drop is grown slowly with a syringe pump (Razel Scientific, R-99) to initiate coalescence, while an AC voltage, $V_{\\text{in}}$ (Hewlett-Packard, HP3325A), is applied across the drops and known circuit elements ($R_k$, $C_k$). \nVoltages $V_1$ and $V_2$ are recorded with a high-speed digitizer (NI PCI-5105, National Instruments) and converted to the time-varying complex impedance of the coalescence cell. \n$Z_{\\text{CR}}$: impedance of the coalescence region (dashed box). \n$Z_t$, $Z_b$: impedances of the fluid-filled nozzles. \n$C_p$: stray capacitance between the nozzles. \n(b-d) Signals for a single saturated aqueous NaCl coalescence versus $\\tau\\equiv t-t_0$. \n(b) The phase angle, $\\Delta\\phi$, between $V_1$ and $V_2$ decreases sharply when the drops touch, which is used to measure $t_0$. \n(c) Capacitance of the coalescence region, $C_{\\text{CR}}$, is roughly constant before and after contact. \n(d) Resistance of the coalescence region, $R_{\\text{CR}}$, after contact.\n}\n\\label{Schematic}\n\\end{figure}\n\nFollowing the AC electrical method developed by Case \\textit{et al.}\\ \\cite{Case2008,Case2009} and used in refs.\\ \\cite{Paulsen2012, Paulsen2011}, I measure the time-varying complex impedance, $Z_{\\text{CR}}$, of two liquid hemispheres while they are coalescing (see Fig.\\ \\ref{Schematic}). \nSalt (NaCl) is added to the drops to make them electrically conductive. \nA high-frequency ($0.6 \\leq f \\leq 10$ MHz), low-amplitude ($V_{\\text{in}} \\leq 2$ V) AC voltage is applied across the drops by gold electrodes that are submerged in the fluid. \nBy simultaneously sampling the voltage below the coalescence cell and the voltage below known passive circuit elements, the impedance of the coalescence cell is determined. \nTwo backgrounds are subtracted: one is measured by bringing the nozzles together, and the second is a small parallel capacitance, $C_p=0.61\\pm 0.12$ pF, that is measured before forming drops on the nozzles. \nThis isolates the impedance of the coalescing drops, $Z_{\\text{CR}}$, which is modeled as a time-varying resistor, $R_{\\text{CR}}$, and capacitor, $C_{\\text{CR}}$, in parallel. \nAt the instant the drops touch, there is a sharp decrease in the phase difference, $\\Delta\\phi$, between the two measured voltages, which indicates the moment of contact, $t_0$, to within 1\/$f$. \n\nExamples of these measured quantities are shown in Fig.\\ \\ref{Schematic}(b-d) as a function of $\\tau \\equiv t-t_0$, which measures time elapsed since the moment of contact, $t_0$. \nMore than $10^4$ points are sampled, thereby capturing a large dynamic range from a single coalescence event. \n\nTo ensure that the applied voltage and the resulting electric fields between the drops do not alter the coalescence dynamics, a variety of checks were performed on the electrical method (see Appendix \\ref{elecSystematic}). \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.15in]{EStat.pdf} \n\\end{center}\n\\caption{\n(Color online) \nConversion between electrical resistance, $R_{\\text{CR}}$, and neck radius, $r_{\\text{min}}$. \n(a) Three axisymmetric models of the coalescence region. \nTop to bottom: two truncated hemispheres are joined with a cylindrical neck of radius $r_{\\text{min}}$, with planar equipotentials (dashed lines) sandwiching the neck; the same geometry without the equipotentials; two hemispheres joined smoothly with a circular arc. \n(b) Electrical resistance versus $r_{\\text{min}}$ calculated numerically for $\\sigma=1$ $\\Omega^{-1}$m$^{-1}$ and $A=2$ mm. \nThe data from all three models are well described by $R_{\\text{CR}}=2\/(\\xi \\sigma r_{\\text{min}})+1\/(\\sigma \\pi A)$ (solid line: Eq.\\ (\\ref{conversion})), where $\\xi=3.62\\pm 0.05$ is a fitting parameter. \nFor small $r_{\\text{min}}$, the data follow $R_{\\text{CR}}=2\/(\\xi \\sigma r_{\\text{min}})$ (dashed line). \n}\n\\label{EStat}\n\\end{figure}\n\nThe conversion between $R_{\\text{CR}}$ and $r_{\\text{min}}$ is geometrical, and was determined numerically using the electrostatics calculation package EStat (FieldCo). \nTo assess the dependance of the conversion on the choice of the model, this conversion was calculated in three different ways, pictured in Fig.\\ \\ref{EStat}(a). \nFirst, the conversion by Case \\textit{et al.}\\ \\cite{Case2008,Case2009} was repeated, in which equipotentials are fixed on two planes that sandwich a cylindrical neck of radius $r_{\\text{min}}$ and height $r_{\\text{min}}^2\/A$, so that the drops and their connecting neck are treated as series contributions to the total resistance. \nThis calculation was compared with a second model with the same geometry but no such restriction on the field lines. \nIn the third model, the shape of the interface is given by a circular arc connecting two hemispheres smoothly. \nAs shown in Fig.\\ \\ref{EStat}(b), the three conversions agree within error bars, and the data are well described by:\n\\begin{equation}\nR_{\\text{CR}} = \\frac{2}{\\xi \\sigma r_{\\text{min}}} + \\frac{1}{\\sigma\\pi A},\n\\label{conversion}\n\\end{equation}\n\n\\noindent where $\\sigma$ is the electrical conductivity of the fluid and the dimensionless constant $\\xi=3.62\\pm 0.05$ is determined empirically. \n\nThe first term in the conversion, $2\/(\\xi \\sigma r_{\\text{min}})$, is twice the resistance of a hemisphere with an opening of radius $r_{\\text{min}}$. \nThe constant term in the conversion, $1\/(\\sigma\\pi A)$, can be understood as coming from the fluid neck itself. \n(This expression is the electrical resistance of a cylinder with radius $r_{\\text{min}}$ and height $r_{\\text{min}}^2\/A$, with equipotentials on its flat faces.) \nOther neck geometries (e.g., an overturned neck shape, which is predicted to occur in the inertial regime \\cite{Eggers2003, Fezzaa2008}) are expected to give the same conversion when $r_{\\text{min}}$ is small, since the dominant term in the resistance comes from the general feature of a conducting hemisphere with an opening of radius $r_{\\text{min}}$. \n\n\n\n\n\n\\subsection{Varying the liquid viscosity}\n\nFor the electrical measurements, the drops were mixtures of glycerol and water, with salt (NaCl) added to make the fluids electrically conductive. \nDe-ionized water was saturated with NaCl at room temperature and mixed with glycerol.\nEach mixture was characterized by measuring its density, surface tension, viscosity, and electrical conductivity. \nDensity was measured by weighing a known volume of fluid. \nSurface tension was measured by matching numerical solutions of the Young-Laplace equation to an image of a pendant drop. \nViscosity was measured with glass capillary viscometers (Cannon-Fenske). \nElectrical conductivity was determined by measuring the electrical impedance of a thin cylindrical channel filled with fluid, using the coalescence cell and measurement circuit. \n\nThe measured fluid parameters are shown in Fig.\\ \\ref{fluidParams}. \nBy changing the volume fraction of glycerol, the liquid viscosity was varied over two decades (from $1.9$ mPa s to $230$ mPa s) while the density and surface tension remained nearly constant, changing by factors of only $1.04$ and $1.6$, respectively. \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.2in]{FluidParams.pdf} \n\\end{center}\n\\caption{\nFluid parameters for glycerol-water-NaCl mixtures used for electrical measurements. \n(a) Mass density, $\\rho$, is approximately constant over the range of mixtures used. \n(b) Surface tension, $\\gamma$, is approximately constant. \n(Aqueous NaCl has $\\gamma=88.5 \\pm 2$ mN\/m, which is higher than for pure water.) \n(c) Viscosity, $\\mu$, varies over a large range. \n(d) AC electrical conductivity, $\\sigma$ (at 1 to 10 MHz), decreases with increasing glycerol concentration. \n(e) AC electrical conductivity as a function of viscosity decreases slightly faster than $\\mu^{-1}$. \nThe low electrical conductivity at high viscosity sets the upper viscosity limit for the electrical method. \n}\n\\label{fluidParams}\n\\end{figure}\n\nAs shown in Fig.\\ \\ref{fluidParams}(e), the electrical conductivity decreases with increasing viscosity, which limits the experimental range of the viscosity of these mixtures with the electrical method. \nFor a fixed, dilute concentration of NaCl, the relationship would obey: $\\sigma\\propto \\mu^{-1}$. \nThis expression comes from combining the Nernst-Einstein law (which relates conductivity to the ionic diffusion coefficients, $D$, at low ionic concentration: $\\sigma\\propto D$) with the Stokes-Einstein equation ($D\\propto \\mu^{-1}$). \nThe conductivity falls off slightly faster than $\\mu^{-1}$, which is consistent with the lower concentration of NaCl in the mixtures as the glycerol fraction is increased.\n(There is another, smaller correction because the mixtures are not at low concentration, which has the opposite effect on the scaling.) \n\n\n\n\\subsection{High-speed imaging}\n\nA high-speed camera (Phantom v12, Vision Research) was used to observe other aspects of the coalescence dynamics, and to measure the neck radius versus time for silicone oils, which are non-conductive. \nThe drops were precisely aligned with respect to the line-of-sight of the camera.\nNeck radii were measured using an edge-locating analysis on the images.\n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.3in]{VideoElectric_rmin.pdf} \n\\end{center}\n\\caption{\nNeck radius versus time for coalescing aqueous NaCl drops ($\\mu=1.88$ mPa s, $\\gamma=88.5$ mN\/m, $\\rho=1180$ kg\/m$^3$). \n(a) Data from the electrical method ($\\bullet$) and high-speed imaging ($\\circ$), where $t_0$ is determined from a simultaneous electrical measurement.\nThe two methods are in good agreement. \nThe electrical data extends to far earlier times. \n(b) Data from the same experiments, on linear-linear axes, showing every camera frame. \nBefore contact, the drop geometry and finite spatial resolution create an apparent neck of radius $110$ $\\mu$m.\nThe earliest imaging point that corresponds to the actual fluid neck is the third frame after $t_0$ ($\\tau=27.0$ $\\mu$s). \n}\n\\label{VideoElectric_rmin}\n\\end{figure}\n\nTo compare electrical measurements with high-speed imaging data, $r_{\\text{min}}$ was measured both electrically and optically for saturated aqueous NaCl drops. \nFor the optical data used in this comparison, $t_0$ was determined from a simultaneous electrical measurement, which was converted to the camera's time-base with a precision of $0.1$ $\\mu$s. \nAs shown in Fig.\\ \\ref{VideoElectric_rmin}(a), the two methods are in good agreement. \nThe comparison serves as a quantitative check on the electrical method, and additionally illustrates the dynamic range gained by the electrical method versus high-speed imaging. \n\nIn the current configuration, the dynamic range of high-speed imaging is determined by spatial resolution, as opposed to timing resolution.\nTo see this, observe that imaging a neck of radius $r_{\\text{min}}$ requires resolving a much smaller feature: the vertical gap between the drops, $r_{\\text{min}}^2\/A$. \nThus, the minimum observable neck radius is set by the condition that $r_{\\text{min}}^2\/A$ is approximately equal to the spatial resolution of the optical setup (i.e., the neck height limits measurements of the neck width). \nFor the experiment in Fig.\\ \\ref{VideoElectric_rmin}, the spatial resolution is 5.3 $\\mu$m\/pixel, so this estimate predicts that $r_{\\text{min}}$ can be seen down to $100$ $\\mu$m, which is consistent with the data. \n(To avoid this optical limitation, one can alternatively image \\textit{through} the neck, as was done in recent drop spreading experiments \\cite{Eddi2013_1}.) \n\nWhen comparing electrical and optical signals, a recent high-speed imaging study of coalescence reported a short delay ($20$ to $60$ $\\mu$s) between the moment of electrical contact (from an electrical trigger for their ultrafast camera) and the first visible motion of the neck \\cite{Thoroddsen2005}. \nThe apparent delay between the electrical signal and visualized motion is now easily accounted for by the period of time when the neck height is smaller than the optical resolution.\nThis explanation is also consistent with those authors' observation that the delay is shorter for smaller drops. \nTo illustrate this point, Fig. \\ref{VideoElectric_rmin}(b) compares electrical and optical measurements of the apparent neck size, $r_{\\text{min}}$. \nIndeed, the early-time optical data give a constant value of $110$ $\\mu$m, corresponding to the radius of the darkened region where the gap between the drops is smaller than the optical resolution. \n\n\n\n\n\n\\section{Purely viscous (Stokes) and inertial regimes}\n\\label{StokesInertialTheory}\n\nFor purely viscous Stokes flow in two dimensions (2D), an exact analytic solution of coalescence was given by Hopper \\cite{Hopper1984,Hopper1990,Hopper1993a,Hopper1993b}. \nThe shape of the fluid interface at any instant during coalescence is an inverse ellipse, given parametrically by: \n\\begin{subequations}\n \\label{Hopper_contour}\n \\begin{align}\n  r(\\theta) & = \\sqrt{2}A\\frac{(1-m^2)(1+m)\\cos \\theta}{\\sqrt{1+m^2}(1+2m \\cos 2\\theta +m^2)}, \\label{Hopper_contour_r} \\\\\n  z(\\theta) & = \\sqrt{2}A\\frac{(1-m^2)(1-m)\\sin \\theta}{\\sqrt{1+m^2}(1+2m \\cos 2\\theta +m^2)}, \\label{Hopper_contour_z}\n \\end{align}\n\\end{subequations}\n\n\\noindent where $0\\leq \\theta <2\\pi$, and the parameter $m$ is mapped to a neck radius by: \n\\begin{equation}\nr_{\\text{min}}=A\\sqrt{2}(1-m)\/\\sqrt{1+m^2}. \n\\label{rmin_vs_m}\n\\end{equation}\n\n\\noindent This family of curves interpolates between two kissing circles ($m=1$) and a single circle ($m=0$). \nFor small neck radius, these shapes limit to: \n\\begin{equation}\n(r^2+z^2)^2=4 A^2 z^2+r_{\\text{min}}^2 r^2.\n\\label{inverse_ellipse}\n\\end{equation}\n\nIn the solution, the neck radius is given as a function of time by:\n\\begin{equation}\n\\frac{\\gamma\\tau}{\\mu A} = \\frac{\\pi \\sqrt{2}}{4} \\int_{m^2}^1 \\frac{ds}{s (1+s)^{1\/2} K(s)},\n\\label{Hopper_exact}\n\\end{equation}\n\n\\noindent where $K(s)$ is the complete elliptic integral of the first kind, and $m$ is related to $r_{\\text{min}}$ by Eq.\\ (\\ref{rmin_vs_m}). \nThe asymptotic behavior of Eq.\\ (\\ref{Hopper_exact}) (in the limit that $r_{\\text{min}}\/A \\rightarrow 0$) is given by the simple expression: \n\\begin{equation}\n\\frac{\\gamma\\tau}{\\mu A} = \\frac{\\pi r_{\\text{min}}}{A} \\left| \\ln\\left(\\frac{r_{\\text{min}}}{8 A}\\right) \\right|^{-1}.\n\\label{Hopper_approx}\n\\end{equation}\n\nThe early-time asymptotic form of 2D Stokes coalescence was extended to three dimensions (3D) by Eggers \\textit{et al.}\\ \\cite{Eggers1999}. \nFor asymptotically small neck radius,\n\\begin{equation}\nr_{\\text{min}} = \\frac{\\gamma\\tau}{\\pi\\mu} \\left|\\ln\\left({\\frac{\\gamma \\tau}{\\mu A}}\\right)\\right|,\n\\label{viscScaling}\n\\end{equation}\n\n\\noindent which they report is a reasonable approximation for $r_{\\text{min}}\\lesssim 0.03 A$. \n\n\n\nFor inertially-dominated flows where the fluid viscosity is negligible, a scaling argument \\cite{Eggers1999} predicted that in this regime,\n\\begin{equation}\nr_{\\text{min}} = D_0 \\left( \\frac{\\gamma A}{\\rho} \\right)^{1\/4} \\tau^{1\/2},\n\\label{invScaling}\n\\end{equation}\n\n\\noindent where $D_0$ is a dimensionless prefactor.\nThis scaling was seen in numerical simulations, which report $D_0=1.62$ \\cite{Eggers2003}. \nHigh-speed imaging experiments \\cite{MenchacaRocha2001, Wu2004, Thoroddsen2005, Bonn2005, Fezzaa2008} and other numerical simulations \\cite{MenchacaRocha2001, Lee2006, Baroudi2014} have also observed this scaling regime and all report $D_0\\approx 1$. \n\n\n\n\n\n\\section{The inertially limited viscous (ILV) regime}\n\\label{ILVregime}\n\n\n\nRecently, Paulsen \\textit{et al.}\\ \\cite{Paulsen2012} showed that there is a third regime of liquid drop coalescence, which had been missed by previous experiments and was unanticipated by theory. \nThe regime arises because the analytical Stokes solution cannot apply at early times, because it violates a simple force-balance when the neck is small. \nNamely, the macroscopic motion of the drops inherent in the Stokes solution requires a larger force than the vanishingly small neck can provide. \nPaulsen \\textit{et al.}\\ \\cite{Paulsen2012} used simulation and experiment to show that at later times when the neck is larger, the Stokes regime is entered. \n\nPaulsen \\textit{et al.}\\ \\cite{Paulsen2012} called this regime the ``inertially limited viscous\" (ILV) regime, because the inertia of the drops prevents the Stokes solution from applying. \nIn the ILV regime, the neck radius is empirically found to follow: \n\\begin{equation}\nr_{\\text{min}} = C_0 \\frac{\\gamma}{\\mu}\\tau.\n\\label{ILV_Scaling}\n\\end{equation}\n\\noindent Previous experiments had observed this linear growth, but incorrectly assumed the drops to be in the Stokes regime \\cite{Bonn2005,Thoroddsen2005,Burton2007,Paulsen2011,Yokota2011}. \n\nPaulsen \\textit{et al.}\\ \\cite{Paulsen2012} used a force-balance argument to predict that for 3D drops, the Stokes regime is entered when: \n\\begin{equation}\nOh \\propto \\left| \\ln\\left(\\frac{1}{8}\\frac{r_{\\text{min}}}{A}\\right)\\right| \\left(\\frac{r_{\\text{min}}}{A}\\right)^{-1\/2}.\n\\label{phase_boundary_3D}\n\\end{equation} \nFig.\\ \\ref{phaseDiagram}(b) shows the phase diagram for liquid drop coalescence in 3D. \nThe ILV regime occupies an increasingly larger portion of the phase-space as $r_{\\text{min}}\/A\\rightarrow 0$.\nThus, surface tension, inertia, and viscosity combine to form the true asymptotic regime of liquid drop coalescence. \n(However, if the drop viscosity is extremely large or small, the range where the ILV regime occurs may be below atomic scales, and so coalescence will start in the Stokes or the inertial regime.) \n\nIn this section, I provide additional measurements and analysis of the ILV and Stokes regimes. \nThese measurements support the new picture of coalescence developed by Paulsen \\textit{et al.}\\ \\cite{Paulsen2012}. \n\n\n\n\n\n\n\n\n\n\\subsection{Change in velocity scaling}\n\nPaulsen \\textit{et al.}\\ \\cite{Paulsen2012} observed that the transition from the ILV regime to the Stokes regime would be accompanied by a change in the macroscopic velocity scaling of the drops. \nTo observe this macroscopic motion, they used a geometry where two pendant drops hang from nozzles and are translated horizontally to initiate contact on their equators. \nPaulsen \\textit{et al.}\\ \\cite{Paulsen2012} measured the velocity of a point on the back of one drop, $v_{\\text{b.o.d.}}$, as a probe of the global motion of the drops, thus identifying the phase boundary between the ILV and Stokes regimes. \nHere, I measure the center-of-mass velocity of each drop, $v_{\\text{c.o.m.}}$, and show that it gives consistent results. \n\nIn the ILV regime, a force balance argument \\cite{Paulsen2012} gives: \n\\begin{equation}\nv_{\\text{c.o.m.}} \\approx \\frac{3\\mu}{4A^3 \\rho} r_{\\text{min}}^2,\n\\label{vcom_ILV}\n\\end{equation}\n\n\\noindent In the Stokes regime, the 2D Stokes solution gives the asymptotic relationship: \n\\begin{equation}\nv_{\\text{c.o.m.}} \\approx \\frac{\\gamma}{2\\pi\\mu} \\left(\\frac{r_{\\text{min}}}{A}\\right) \\left|\\ln\\left(\\frac{1}{8}\\frac{r_{\\text{min}}}{A}\\right)\\right|,\n\\label{vcom_Stokes}\n\\end{equation}\n\\noindent which should apply for 3D drops as well \\cite{Eggers1999}.\n\nHigh-speed movies of the coalescing drops are analyzed to give the position of the center-of-mass of one drop, which is numerically differentiated to give $v_{\\text{c.o.m.}}$, and averaged to suppress noise. \n(Because the movies only give the planar drop contour, I calculate the center-of-mass of the shape that is given by revolving the contour of the bottom half of one drop around the axis passing through the center of both drops.) \nThe neck radius is measured directly from the same movie. \n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=2.8in]{Vcom_vs_rmin.pdf}\n\\end{center}\n\\caption{\n(Color online) \nILV-to-Stokes crossover. \n(a) Rescaled center-of-mass velocity of the drops, $v_{\\text{c.o.m.}}\\mu\/\\gamma$, versus $r_{\\text{min}}\/A$ at several viscosities. \nSolid line: asymptotic result from 2D Stokes theory, Eq.\\ (\\ref{vcom_Stokes}). \nThe data show super-linear growth at early times and merge onto the Stokes solution at late times. \nHigher-viscosity drops enter the Stokes regime at smaller neck radius. \n(b) The center-of-mass motion follows the motion of the back of one drop, here shown for $Oh=3.1$. \n}\n\\label{Vcom_vs_rmin}\n\\end{figure}\n\nFigure \\ref{Vcom_vs_rmin}(a) shows $v_{\\text{c.o.m.}}$ rescaled by the viscous-capillary velocity, $\\gamma\/\\mu$, versus the non-dimensional neck radius, $r_{\\text{min}}\/A$, for several viscosities. \nThe data capture both an early dynamics where the global drop velocity is growing approximately as $r_{\\text{min}}^2$ (as predicted for the ILV regime by Eq.\\ (\\ref{vcom_ILV})), and a late dynamics, where the data merge onto a master curve that is consistent with the Stokes theory, Eq.\\ (\\ref{vcom_Stokes}). \nThe higher the fluid viscosity, the earlier the transition into the Stokes regime. \nIn Fig.\\ \\ref{Vcom_vs_rmin}(b), $v_{\\text{c.o.m.}}$ is shown for one of the viscosities along with $v_{\\text{b.o.d.}}$ obtained from the same movie. \nThe two measurements are in good agreement. \nThis crossover in global drop motion marks the phase boundary between the ILV regime and the Stokes regime, which was reported in ref.\\ \\cite{Paulsen2012}, and is plotted in Fig.\\ \\ref{phaseDiagram}(b). \n\n\n\n\\subsection{Neck shapes}\n\nThe shape of the fluid neck connecting the coalescing drops offers another means of identifying the Stokes regime from the ILV regime. \nThe neck shapes were compared by Paulsen \\textit{et al.}\\ \\cite{Paulsen2012}, and a more detailed comparison is provided here. \n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=3.3in]{DropProfileCurvature.pdf}\n\\end{center}\n\\caption{\n(Color online) \nNeck shape versus $r_{\\text{min}}\/A$ and viscosity. \n(a-c) Neck shape for high viscosity (a) ($\\mu=58600$ mPa s, $Oh=370$) and intermediate viscosity (b,c) ($\\mu=96.6$ mPa s, $Oh=0.62$). \nAt high viscosity (a), the data agree with the exact Stokes-flow shapes (solid lines: Eq.\\ (\\ref{Hopper_contour})). \nAt intermediate viscosity (b), the neck is much broader, and the Stokes theory is a poor fit to the data. \nInstead, the data is well described by two spheres joined smoothly with a parabolic neck (c). \n(d) Neck shape similarity solutions versus rescaled coordinates, $\\tilde{r} \\equiv r\/r_{\\text{min}}$, $\\tilde{z} \\equiv z\/(r_{\\text{min}}^2\/A)$. \nDotted line: parabolic neck connected to spherical drops, Eq.\\ (\\ref{simsol_Parabola}). \nSolid line: Stokes solution, Eq.\\ (\\ref{simsol_Hopper}). \n(e) Dimensionless radius of curvature at neck minimum, $1\/\\kappa A$, versus $r_{\\text{min}}\/A$. \nHigh-viscosity data ($\\bullet$ $Oh=370$) agree with the Stokes theory (dashed line), which is approximated by Eq.\\ (\\ref{neck_curve}) with $C=1\/4$ at early times (solid line). \nIntermediate-viscosity data ($\\circ$ $Oh=0.62$) follow the result for a parabolic neck (dotted line: Eq.\\ (\\ref{neck_curve}) with $C=32\/27$). \n(f) Curvature scaling prefactor, $C$, measured at fixed radius ($0.1<r_{\\text{min}}\/A<0.2$), versus $Oh$.\nAs viscosity increases, the data transition from the value for a parabolic neck (dotted line: $32\/27$) to the Stokes theory (solid line: $1\/4$). \n}\n\\label{DropProfileFig}\n\\end{figure}\n\nIn Fig.\\ \\ref{DropProfileFig}(a) and (b), the exact shapes, Eq.\\ (\\ref{Hopper_contour}), are compared with experiments at $Oh=370$ and $Oh=0.62$.\nAt each viscosity, the neck shapes are shown at three different times. \nTo demonstrate that the regime can be identified by shape alone (without a knowledge of the time-dependance), I plot the shape that matches the minimum neck radius, $r_{\\text{min}}$, of the experimental data. \nThe Stokes solution agrees well with the high-viscosity data, where the drops have transitioned into the Stokes regime. \nHowever, it clearly fails to describe the shapes at $Oh=0.62$, where the drops are in the ILV regime, as shown in Fig.\\ \\ref{DropProfileFig}(b). \nIn this regime, the neck is broader, perhaps because the drops have not translated towards each other appreciably. \nInstead, the neck shapes in this regime can be described by two kissing spheres joined smoothly by a parabolic neck, as shown in Fig.\\ \\ref{DropProfileFig}(c). \n(The parabola is uniquely determined by specifying $r_{\\text{min}}$ and requiring that the parabolic region joins the two spheres continuously and with a continuous first derivative.)\n\nOn intermediate scales that are larger than the neck but smaller than the drops, the spherical drops are well-approximated by parabolas: $r=\\sqrt{2Az}$ for $r_{\\text{min}}\\ll r \\ll A$. \nA change of variables is made by rescaling the radial direction by the neck radius, $\\tilde{r} \\equiv r\/r_{\\text{min}}$, and the axial direction by the neck height, $\\tilde{z} \\equiv z\/(r_{\\text{min}}^2\/A)$. \nThis transformation collapses the profile onto itself: \n\\begin{equation}\n\\tilde{r}(\\tilde{z}) =\n\\begin{cases} \n1+\\frac{27}{64}\\tilde{z}^2, & \\tilde{z}< \\frac{8}{9}; \\\\\n\\sqrt{2 \\tilde{z}}, & \\tilde{z}\\geq \\frac{8}{9}. \\\\\n\\end{cases}\n\\label{simsol_Parabola}\n\\end{equation}\n\n\\noindent Thus, the parabolic-neck geometry is a similarity solution with radial scale $r_{\\text{min}}$ and axial scale $r_{\\text{min}}^2\/A$. \nThe solution is plotted in Fig.\\ \\ref{DropProfileFig}(d), with a point at $(\\tilde{z},\\tilde{r})=(8\/9,4\/3)$ marking where the neck merges onto the drops.\n(The same rescaled coordinates, $r\/r_{\\text{min}}$ and $z\/(r_{\\text{min}}^2\/A)$, were found to collapse the neck shapes in the inertial regime, for the case of neighboring spherical drops deposited on a substrate with a 90$^{\\circ}$ contact angle \\cite{Eddi2013_2}.) \n\nThe drop shapes in the Stokes regime also form a similarity solution near the neck with the same scaling. \nAs noted in ref.\\ \\cite{Eggers1999}, expansion of the drop contour near the neck for $z\\ll r_{\\text{min}}\\ll A$ (using Eq.\\ (\\ref{Hopper_contour}) or Eq.\\ (\\ref{inverse_ellipse})) gives $r(z)=(\\frac{1}{2}r_{\\text{min}}^2+\\sqrt{(\\frac{1}{2}r_{\\text{min}}^2)^2+4A^2z^2})^{1\/2}$. \nRescaling this shape yields:\n\\begin{equation}\n\\tilde{r}(\\tilde{z}) = \\left(\\frac{1}{2}+\\sqrt{\\frac{1}{4}+4\\tilde{z}^2}\\right)^{1\/2},\n\\label{simsol_Hopper}\n\\end{equation}\n\n\\noindent which is plotted in Fig.\\ \\ref{DropProfileFig}(d). \n\nAlthough I have only identified the self-similarity of the drop \\textit{shapes} during coalescence, it is possible that the flows share this self-similarity. \nIf so, then radial flows will scale with the neck radius, $r_{\\text{min}}$, and axial flows will scale with the neck height, $r_{\\text{min}}^2\/A$, in both the ILV and Stokes regimes. \nThis further solidifies a claim argued in ref.\\ \\cite{Paulsen2011} and in section \\ref{VIcrossover} that the flow scale in the axial direction is given by the neck height, $r_{\\text{min}}^2\/A$. \n\nThe interfacial curvature in the $(z,r)$ plane at the neck minimum, $\\kappa$, is a scalar quantity that distinguishes between the asymptotic neck shapes in the ILV and Stokes regimes. \nIn both regimes, the similarity solutions obey: \n\\begin{equation}\n\\frac{1}{\\kappa A}=C \\left(\\frac{r_{\\text{min}}}{A}\\right)^3,\n\\label{neck_curve}\n\\end{equation}\n \n\\noindent where $C=\\frac{32}{27}\\approx 1.19$ in the ILV regime, and $C=\\frac{1}{4}$ in the Stokes regime. \nThese predictions are in good agreement with the data shown in Fig.\\ \\ref{DropProfileFig}(e) for the two regimes. \n\nFigure \\ref{DropProfileFig}(f) shows $C$ at fixed radius ($0.1<r_{\\text{min}}\/A<0.2$) as a function of dimensionless viscosity, $Oh$. \nNear $Oh=1.5$, there is a transition between the values predicted for the Stokes and ILV regimes, marking the phase boundary at that radius. \n\n\n\n\\subsection{Effect of drop boundary condition}\n\nIn ref.\\ \\cite{Paulsen2012} and in this work, it has been shown that the Stokes and ILV regimes have different macroscopic drop motion. \nOne natural question is whether the boundary condition on the drops can influence this motion and determine the coalescence regime (e.g., is the Stokes regime precluded if the drops are fixed to rigid objects?). \nIn terms of the coalescence singularity, the crucial question is whether the boundary condition can affect the dynamics in the limit that $r_{\\text{min}}\/A\\rightarrow 0$. \n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=3.2in]{NozzleGeomCompare.pdf}\n\\end{center}\n\\caption{\n(Color online) Effect of drop boundary conditions.\n(a) High-viscosity coalescence (silicone oil with $\\mu=58600$ mPa s, $Oh=370$) for vertically-aligned hemispherical drops formed on nozzles ($\\circ$) and for hanging pendant drops ($\\triangle$). \nAt early times, the neck growth does not depend on the boundary condition. \nThe data follow the Stokes solution (solid line: Eq.\\ (\\ref{Hopper_exact})). \n(b) Intermediate-viscosity coalescence in these two drop geometries (glycerol-water-NaCl with $\\mu=230$ mPa s ($\\circ$) and $\\mu=215$ mPa s ($\\triangle$); $Oh\\approx 1$).\nThe neck growth does not depend on the boundary condition. \nThe data follow the ILV scaling (dashed line: Eq.\\ (\\ref{ILV_Scaling})). \n}\n\\label{NozzleGeomCompare}\n\\end{figure}\n\nTo address this issue, I compare measurements of $r_{\\text{min}}$ under two contrasting boundary conditions: (i) the drops are hemispheres attached to fixed nozzles, and (ii) the drops are hanging and are coalesced at their equators, so that they can freely translate towards each other as they coalesce. \nI measure $r_{\\text{min}}$ with both boundary conditions at high viscosity and intermediate viscosity. \n\nFigure \\ref{NozzleGeomCompare}(a) compares high viscosity data with the 2D Stokes theory, Eq.\\ (\\ref{Hopper_exact}). \n(While the 2D and 3D theories match at early times, the 3D case does not make a prediction for $r_{\\text{min}}> 0.03 A$, where all of the data lie. \nTherefore, following ref.\\ \\cite{Paulsen2012}, the data are compared with the 2D exact analytic solution.) \nThe data follow the theory for small $r_{\\text{min}}\/A$; only at later times do the curves begin to depart from each other. \nFigure \\ref{NozzleGeomCompare}(b) shows that for intermediate-viscosity drops, the boundary condition has a negligible effect on the dynamics, and the data matches the neck scaling for the ILV regime, Eq.\\ (\\ref{ILV_Scaling}). \nThus, the phase diagram shown in Fig.\\ \\ref{phaseDiagram}(b) applies to coalescing drops that are fixed or free. \n\n\n\n\n\n\\section{Viscous-to-inertial crossover}\n\\label{VIcrossover}\n\nThus far, I have reported coalescence measurements in the Stokes and the ILV regimes, and I have observed the ILV-to-Stokes crossover by measuring the macroscopic motion of the drops. \nThe remaining component of the coalescence phase diagram is the viscous-to-inertial crossover (from the ILV regime to the inertial regime). \n\nRecently, Paulsen \\textit{et al.}\\ \\cite{Paulsen2011} used an ultrafast electrical method to measure this crossover for salt-water drops, and reported a major discrepancy with the theory \\cite{Eggers1999,Eggers2003}. \nWhereas the theory predicts a crossover time between these regimes of $t_c \\approx 0.7$ ns, the experiments show $t_c \\approx 2$ $\\mu$s. \nIn terms of the neck size, the crossover radius was predicted to be $r_c \\approx 30$ nm, whereas experiment showed $r_c \\approx 20$ $\\mu$m.\n\nTo investigate this discrepancy, experiments were carried out where the liquid viscosity was varied over a large range \\cite{Paulsen2011}. \nThe data was found to be consistent with a newly proposed Reynolds number for coalescence, which is based on a smaller length scale for the dominant flow gradients given by the neck height, $L=r_{\\text{min}}^2\/A$. \nHere, I provide additional measurements and analysis of the viscous-to-inertial crossover, which support the conclusions of ref.\\ \\cite{Paulsen2011}. \n\n\n\n\n\\subsection{Collapse of electrical data}\n\\label{PrelimComp}\n\n\n\n\n\nThe inset to Fig.\\ \\ref{muCollapse} shows $r_{\\text{min}}$ versus time for 4 viscosities, ranging from $1.9$ to $82$ mPa s, which were measured electrically. \nIn ref.\\ \\cite{Paulsen2011}, these data were rescaled to fall onto a master plot by rescaling the vertical and horizontal axes with free parameters, $r_c$ and $\\tau_c$, at each viscosity to produce the best collapse. \nI perform a different analysis here, in order to demonstrate that the results are not sensitive to the particular way in which the data collapse is obtained. \n\nIn ref.\\ \\cite{Paulsen2011}, after collapsing the data, it was noted that all of the data followed the simple interpolation:\n\\begin{equation}\n\\frac{r_{\\text{min}}}{r_c}=2 \\left(\\frac{1}{\\tau\/\\tau_c} + \\frac{1}{\\sqrt{\\tau\/\\tau_c}}\\right)^{-1}.\n\\label{interpolate}\n\\end{equation}\n\n\\noindent Here, I start with Eq.\\ (\\ref{interpolate}) and use it to collapse the data. \nFor each viscosity, Eq.\\ (\\ref{interpolate}) is fit to the data, where $r_c$ and $\\tau_c$ are fitting parameters. \nThe data is then rescaled by the $r_c$ and $\\tau_c$.\n\n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.2in]{muCollapse.pdf}\n\\end{center}\n\\caption{\n(Color online) \n\\textit{Inset}: Neck radius versus time for glycerol-water-NaCl mixtures of different viscosities, from $1.9$ to $82$ mPa s. \nAt each viscosity, data from 5 or more coalescence events are logarithmically binned and averaged. \n\\textit{Main}: The data is collapsed by rescaling the x- and y-axes. \nRescaling parameters $\\tau_c$ and $r_c$ are obtained for each viscosity by fitting the data to Eq.\\ (\\ref{interpolate}) (solid line).\nThe collapsed data and the fit exhibit asymptotic behavior of $2\\tau\/\\tau_c$ (dotted line) at early times and $2\\sqrt{\\tau\/\\tau_c}$ (dashed line) at late times. \n} \n\\label{muCollapse}\n\\end{figure}\n\nFigure \\ref{muCollapse} shows the collapsed data, which fall cleanly onto a single curve given by Eq.\\ (\\ref{interpolate}). \nThe scaling parameters, $r_c$ and $\\tau_c$, determine the coefficients for the early- and late-time scaling laws, $C_0$ and $D_0$ (defined by eqns.\\ \\ref{ILV_Scaling} and \\ref{invScaling}, respectively). \nFigure \\ref{rcvsOh}(a) and (b) shows these coefficients as a function of dimensionless viscosity, $Oh$. \nThe ILV scaling prefactor, $C_0$, is of order 1 across the entire range of viscosity (although there is a slight increase in $C_0$ as the viscosity is increased). \nThe inertial scaling prefactor, $D_0$, is in good agreement with the value from numerical simulations \\cite{Eggers2003}, $D_0=1.62$. \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center}\n\\includegraphics[width=3.4in]{rcOh.pdf} \n\\end{center}\n\\caption{\n(a,b) Measured dimensionless scaling-law prefactors, $C_0$ and $D_0$, versus $Oh$. \nIn (a), the dashed line is $C_0=1$. \nIn (b), the dashed line is the value from simulation \\cite{Eggers2003}: $D_0=1.62$. \n(c) Rescaled viscous-to-inertial crossover time versus $Oh$. \nThe dashed line shows $\\tau_c\/\\tau_{\\text{v}} = Oh^2$ (where $\\tau_{\\text{v}}= \\mu A\/\\gamma$ is the viscous timescale), as predicted in the literature \\cite{Eggers1999, Eggers2003}. \nClearly this is a poor description of the data. \nThe crossover radius proposed by ref.\\ \\cite{Paulsen2011} (with $\\tau_c\/\\tau_{\\text{v}} \\propto Oh$) is consistent with the data (solid line: Eq.\\ (\\ref{ourtc})). \n(d) Rescaled viscous-to-inertial crossover radius versus $Oh$.\nThe dashed line shows $r_c\/A = Oh^2$, which was proposed in the literature \\cite{Eggers1999, Eggers2003}.\nThis fails to capture the data. \nThe crossover radius proposed in this work describes the data well, with $r_c\/A \\propto Oh$ (solid line: Eq.\\ (\\ref{rcformula})). \nIn (a-d), the error bars are determined by the fits to Eq.\\ (\\ref{interpolate}). \n} \n\\label{rcvsOh}\n\\end{figure}\n\nFigure \\ref{rcvsOh}(c) shows the dimensionless crossover time, $\\tau_c\/\\tau_{\\text{v}}$, as a function of $Oh$ (where $\\tau_{\\text{v}}= \\mu A\/\\gamma$ is the viscous timescale). \nClearly, the accepted formula for the crossover time, $\\tau_c\/\\tau_{\\text{v}} \\approx Oh^2$, does not agree with the data. \nThe measurements are better described by a linear dependence on $Oh$. \n\nThe discrepancy between theory and experiment is also evident in the dimensionless crossover radius, $r_c\/A$, versus $Oh$, shown in Fig.\\ \\ref{rcvsOh}(d). \nThe predicted crossover radius is $r_c\/A \\approx Oh^2$, whereas the data follow $r_c\/A \\approx Oh$. \nThis suggests that the conventional Reynolds number for coalescence, $Re = \\rho \\gamma r_{\\text{min}}\/\\mu^2$, is wrong. \n\n\n\n\\subsection{Reynolds number for coalescence}\n\nThe viscous-to-inertial crossover can be estimated by the condition that the dimensionless Reynolds number for the flows, $Re=\\rho U L\/\\mu$, is of order unity (where $U$ and $L$ are characteristic velocity- and length-scales in the flows, respectively). \nAs was argued in ref.\\ \\cite{Paulsen2011}, the dominant flows in the viscous-to-inertial crossover correspond to a different Reynolds number than the one used in the literature \\cite{Eggers1999, Eggers2003, Wu2004, Yao2005, Thoroddsen2005, Bonn2005}. \nInstead of the conventionally used length scale given by the neck radius, $L=r_{\\text{min}}$, a much smaller length scale---the neck height, $r_{\\text{min}}^2\/A$---describes the size of the flow gradients. \n\nPaulsen \\textit{et al.}\\ \\cite{Paulsen2011} gave an estimate for the Reynolds number coming from the inertial side of the crossover. \nThey found that the crossover time, $\\tau_c$, is given by: \n\\begin{equation}\n\\frac{\\tau_c}{\\tau_{\\text{v}}} \\approx \\frac{64}{D_0^6} \\left(\\frac{\\mu}{\\sqrt{\\rho \\gamma A}}\\right) = \\frac{64}{D_0^6} \\mbox{ } Oh,\n\\label{ourtc}\n\\end{equation}\n\\noindent which is written here using the viscous timescale, $\\tau_{\\text{v}}$, and the Ohnesorge number. \nFigure \\ref{rcvsOh}(c) shows that this prediction is consistent with the crossover times measured in this work. \n\n\n\n\n\n\n\nA similar argument can be made coming from the viscous side of the crossover, which is presented here. \nIn the early (ILV) regime, the characteristic speed of the flows is $U=\\gamma\/\\mu$, and the characteristic length-scale is $L=r_{\\text{min}}^2\/(2A)$, since liquid from each drop moves in to advance the neck. \nUsing these scales, the Reynolds number is: $Re=\\rho\\gamma r_{\\text{min}}^2\/(2 A \\mu^2)$. \n\\noindent The dimensionless crossover radius, $r_c\/A$, is obtained by setting $Re=1$: \n\\begin{equation}\n\\frac{r_c}{A}\\approx \\sqrt{2}\\left(\\frac{\\mu}{\\sqrt{\\rho \\gamma A}}\\right)=\\sqrt{2}\\mbox{ }Oh.\n\\label{rcformula}\n\\end{equation}\n\n\\noindent Figure \\ref{rcvsOh}(d) shows that this prediction gives excellent agreement with the data\n\nIn Appendix \\ref{prevCross}, I compare the calculated crossover time, Eq.\\ (\\ref{ourtc}), with previous measurements of the viscous-to-inertial crossover in the literature. \n\n\n\n\\subsection{High-speed imaging collapse}\n\nHigh-speed videos of coalescence show that these results also capture the dependance of the crossover on surface tension. \nI coalesce glycerol-water-NaCl mixtures with viscosities ranging from 1.9 to 230 mPa s, and silicone oils with viscosities ranging from 0.82 to 97 mPa s. \n(The silicone oils are electrically insulating, and therefore cannot be measured with the electrical method.) \nUsing these liquids, the surface tension is varied by a factor of 5.\nThe liquids have $Oh<1$ so that the behavior should be described by the ILV regime and the inertial regime, but not the Stokes regime. \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.1in]{videoCollapse.pdf} \n\\end{center}\n\\caption{\n(Color online)\nHigh-speed imaging of coalescence. \n\\textit{Inset:} Neck radius versus time for glycerol-water-NaCl mixtures ($\\mu=1.9$, 30, and 230 mPa s) and silicone oils ($\\mu=0.82$ and 97 mPa s). \nOther parameters are listed in the legend. \n\\textit{Main:} The data is collapsed by rescaling the axes with the crossover radius, $r_c$ (calculated with Eq.\\ (\\ref{rcformula})), and the crossover time, $\\tau_c$ (calculated with Eq.\\ (\\ref{ourtc})). \nThe rescaled data are consistent with Eq.\\ (\\ref{interpolate}) (solid line). \n} \n\\label{videoCollapse}\n\\end{figure}\n\nThe inset of Fig.\\ \\ref{videoCollapse} shows $r_{\\text{min}}$ versus $\\tau$ for these liquids. \nWhen the axes are rescaled with $r_c$ (given by Eq.\\ (\\ref{rcformula})) and $\\tau_c$ (given by Eq.\\ (\\ref{ourtc})), the data collapse to a master curve, shown in Fig.\\ \\ref{videoCollapse}. \nThe collapsed data follow Eq.\\ (\\ref{interpolate}), and therefore fall on the electrical data collapse, Fig.\\ \\ref{muCollapse}. \nThese experiments further solidify the new phase diagram for coalescence, shown in Fig.\\ \\ref{phaseDiagram}(b). \n\n\n\n\n\n\\section{Dynamics of drops during approach}\n\\label{approach}\n\n\\subsection{Drop deformation}\n\nThe experiments in this work were performed at ambient air pressure. \nBecause the drops approach at finite speed, they can be deformed by the viscous stresses in the air layer between them \\cite{Neitzel2002}. \nThis deformation could affect the subsequent coalescence dynamics. \nPrevious experiments \\cite{Case2008,Case2009} using the same electrical method suggest that deformation may be present for approach speeds as low as $10^{-4}$ m\/s. \n\nHere, aqueous NaCl drops are coalesced in air at an approach speed that is varied over $7$ orders of magnitude down to $17$ nm\/s, to examine the effects of the ambient gas during approach. \nTo achieve constant approach-speeds lower than $10^{-3}$ m\/s, a variable-speed syringe pump was used with a wide range of syringe sizes.\nThe approach speed, $U_{\\text{app}}$, was calculated based on the geometry and the flow rate. \nThe coalescence cell was fixed to a vibration-isolation table to suppress disturbances on the drops. \nFor high approach-speeds, a gravity-fed system was used: the bottom drop was fed by a reservoir held at a variable height above the coalescence cell, so that hydrostatic pressure caused the bottom drop to grow and impact the top drop. \nFor the gravity-fed system, $U_{\\text{app}}$ was measured directly with a high-speed camera. \n\nFigure \\ref{RCRva}(a) shows an image taken within one frame of $t_0$ for $U_{\\text{app}}=8.8\\times 10^{-5}$ m\/s. \nThe drops appear to be undeformed at the moment of contact. \nAt much higher approach-speed, the drops visibly deform before they merge, as shown in Fig.\\ \\ref{RCRva}(b), for $U_{\\text{app}}=3.3\\times 10^{-2}$ m\/s. \nThis transient non-coalescence is due to the pressure provided by the lubricating air layer between the drops. \nAlthough the drops appear undeformed in the low approach-speed case shown in Fig.\\ \\ref{RCRva}(a), the image does not rule out the possibility of a small flattened region at the drop tips. \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.2in]{dropsRCRva.pdf} \n\\end{center}\n\\caption{\nOptical and electrical measurements of coalescence at low and high approach-speed. \n(a,b) Visual indications of drop deformation. \nDrops are shown within one frame of $t_0$ at low approach-speed: (a) $U_{\\text{app}}=8.8\\times 10^{-5}$ m\/s, and high approach-speed: (b) $U_{\\text{app}}=3.3\\times 10^{-2}$ m\/s. \nAt low approach-speed, the drops appear to coalesce as undeformed spheres, whereas the high approach-speed drops are flattened. \n(c,d) Electrical measurements corresponding to the experiments shown in (a,b). \n$R_{\\text{CR}}-R_0$ versus $\\tau$, where $R_0=1\/(\\sigma \\pi A)$. \nAt low approach-speed (c), the resistance follows $\\tau^{-1}$ (ILV scaling, dashed line) at early times and $\\tau^{-1\/2}$ (inertial scaling, dotted line) at late times. \nAt high approach-speed (d), the resistance follows $\\tau^{-0.72}$ at early times (solid line). \n}\n\\label{RCRva}\n\\end{figure}\n\nThe electrical method was used to access these small scales. \nFigure \\ref{RCRva}(c) shows electrical measurements of the coalescing drops for the low approach-speed case. \nThe data follow the behavior shown in earlier sections of this work (e.g., Fig.\\ \\ref{Schematic}(d)). \nHowever, for the high approach-speed case, the electrical measurements are qualitatively different, as shown in Fig.\\ \\ref{RCRva}(d). \nAt early times, the resistance appears to follow an approximate power-law with a scaling exponent of $-0.72$.\nAt late times, there is an abrupt crossover out of this scaling.\n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=2.7in]{tauc_Cfinal_Uapp.pdf} \n\\end{center}\n\\caption{\n(a) Crossover time between the early and late electrical-resistance scalings versus approach speed. \nThe crossover time is constant for $U_{\\text{app}}<U_{\\text{app}}^*$, where $U_{\\text{app}}^*=(3 \\pm 1)\\times 10^{-4}$ m\/s. \nWhen $U_{\\text{app}}>U_{\\text{app}}^*$ (shaded region), $\\tau_c$ depends on the approach speed, and is delayed as $U_{\\text{app}}$ increases. \n(b) $C_{\\text{final}}$ versus approach speed. \n$C_{\\text{final}}$ is constant for $U_{\\text{app}}<U_{\\text{app}}^*$. \nFor $U_{\\text{app}}>U_{\\text{app}}^*$ (shaded region), $C_{\\text{final}}$ increases with $U_{\\text{app}}$, consistent with an increase in the flattening of the drops before they touch. \nThe data are averaged over 800 samples within the final 100 $\\mu$s before $t_0$, and $V_{\\text{in}} \\leq 275$ mV.  \n} \n\\label{Uapp}\n\\end{figure}\n\nA crossover time, $\\tau_c$, is measured by fitting the early- and late-time data to separate power-laws and determining the point of intersection of the fits. \n(This criteria is equivalent to fitting to Eq.\\ (\\ref{interpolate}) if the two scalings are the ILV and inertial scalings.) \nFigure \\ref{Uapp}(a) shows $\\tau_c$ versus approach speed. \nThe crossover time is insensitive to the drop approach-speed for $U_{\\text{app}}<3 \\times 10^{-4}$ m\/s. \nFor $U_{\\text{app}}>3 \\times 10^{-4}$ m\/s, the crossover time increases approximately linearly with $U_{\\text{app}}$, which is correlated with an increase in flattening in the high-speed videos. \nA threshold approach-speed, $U_{\\text{app}}^*=(3 \\pm 1)\\times 10^{-4}$ m\/s, separates the two behaviors. \n\nThe capacitance of the drops at the moment of contact, $C_{\\text{final}}\\equiv C_{\\text{CR}}(\\tau=0)$, should be sensitive to the amount of drop deformation as well.\nIn particular, $C_{\\text{final}}$ should grow with the area of the deformed region.\nFig.\\ \\ref{Uapp}(b) shows $C_{\\text{final}}$ versus approach speed. \nThe capacitance shows two behaviors, which fall on either side of the threshold approach-speed, $U_{\\text{app}}^*$. \nAt high approach-speed ($U_{\\text{app}}>U_{\\text{app}}^*$), $C_{\\text{final}}$ increases with $U_{\\text{app}}$ as the drops are increasingly deformed. \nAt low approach-speed ($U_{\\text{app}}<U_{\\text{app}}^*$), $C_{\\text{final}}$ is independent of $U_{\\text{app}}$, which is consistent with a picture where the drops are undeformed. \n\nThe crossover time and capacitance measurements suggest that drop deformation is absent at low approach-speed, since in the flattening scenario, one expects the amount of flattening to change with $U_{\\text{app}}$ \\cite{Law2007}.\nIn general, $U_{\\text{app}}^*$ may depend on the drop size, surface tension, the ambient fluid viscosity, and the drop viscosity. \n\n\n\\subsection{Inception of coalescence}\n\nTo probe the dynamics up to the moment of contact, I measure the capacitance of the coalescence region, $C_{\\text{CR}}$, during approach at low speed ($U_{\\text{app}}<U_{\\text{app}}^*$). \nThe separation of the drop surfaces is denoted by $z$ (see Fig.\\ \\ref{CfinalVa}(c)). \nThe mutual capacitance of the two hemispherical drops at small separation ($z\\ll A$) should be comparable to the capacitance of two conducting spheres, since the surfaces in close proximity contribute the most charge. \nThe latter arrangement can be solved exactly as a series expansion. \nFor small $z\/A$, the capacitance is given to a good approximation \\cite{Love1979} by: \n\\begin{equation}\nC_{\\text{spheres}}(z)\\approx \\pi\\epsilon_0 A \\left[\\ln\\left(\\frac{A}{z}\\right)+2.54\\right].\n\\label{Ceq}\n\\end{equation}\n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.2in]{CversusMinusTau.pdf} \n\\end{center}\n\\caption{\n(Color online)\nDrop capacitance before coalescence for drops approaching at speed $U_{\\text{app}}=8.8\\times 10^{-5}$ m\/s $<U_{\\text{app}}^*$. \n(a) Capacitance of the coalescence region, $C_{\\text{CR}}$, versus time remaining until coalescence, $-\\tau$, with $V_{\\text{in}}=275$ mV and $f=10$ MHz. \nThe drops initiate contact at finite separation, shown by the excellent fit to Eq.\\ (\\ref{Ceq}) with $z=-U_{\\text{app}} \\tau+z_0$ using $z_0=160$ nm (solid line). \nFor comparison, the same equation is plotted with $z_0=0$ (dashed line). \nThe data is averaged and logarithmically binned over 12 experiments.\n(b) $C_{\\text{final}}$ versus applied voltage, $V_{\\text{in}}$. \nAt high voltage, the data is consistent with coalescence initiating when the peak electric field between the drops reaches a threshold value (dotted line: Eq.\\ (\\ref{Ceq}) with $z_0=V_{\\text{in}}\/E_{\\text{thresh}}$ using $E_{\\text{thresh}}=1.2$ MV\/m). \nAt lower voltage, $C_{\\text{final}}$ is consistent with a constant value that is independent of $V_{\\text{in}}$ (solid line: $C_{\\text{final}}=0.63 \\pm 0.05$ pF). \nEach data point is averaged over 800 samples within the final 100 $\\mu$s before $t_0$. \nThree measurements were made at each voltage to show the run-to-run variation. \n(c) Approach geometry: two drops of radius $A$ separated by distance $z$. \n} \n\\label{CfinalVa}\n\\end{figure}\n\nFigure \\ref{CfinalVa}(a) shows the capacitance of the coalescence region, $C_{\\text{CR}}$, measured as a function of the time remaining until coalescence, $-\\tau$, for aqueous NaCl drops approaching at low speed. \nThe capacitance grows logarithmically with $-\\tau$ at first but does not diverge at $\\tau=0$. \nThis behavior is consistent with the drops initiating contact at a finite separation, $z_0$. \nPlugging $z=-U_{\\text{app}} \\tau+z_0$ into Eq.\\ (\\ref{Ceq}) and using $z_0=160$ nm gives excellent agreement with the data. \n\n\nDespite the low applied voltage, the electric field between the drops can be very strong when the drop separation is small. \nThis is apparent when estimating the peak magnitude of the electric field between the drops during one AC cycle as $E_{\\text{max}}\\approx V_{\\text{in}}\/z$. \nOne expects the large electric field to attract the drops towards each other when they are very close, thus promoting coalescence and distorting the drop shapes before contact. \n\nTo test the effect of the applied voltage on the initiation of coalescence, $C_{\\text{final}}$ was measured as a function of applied voltage, $V_{\\text{in}}$. \nFigure \\ref{CfinalVa}(b) shows the results for $V_{\\text{in}}$ ranging from 2.5 mV to 5 V.\nFor $V_{\\text{in}}>0.3$ V, the data are consistent with the drops forming a connecting neck when the intervening electric field exceeds a threshold value, $E_{\\text{thresh}}$. \nThe data are fit well by $E_{\\text{thresh}}=1.2 \\pm 0.2$ MV\/m, using Eq.\\ (\\ref{Ceq}) for the capacitance of the drops as a function of the final separation, $z_0$, and substituting $z_0\\approx V_{\\text{in}}\/E_{\\text{thresh}}$. \nWhile this value is only slightly smaller than the approximate dielectric strength of air at large distances (3 MV\/m), the dielectric strength of air at these short distances is much greater \\cite{Townsend1915}.\n\nHaving argued that dielectric breakdown does not occur, I now address whether the applied voltage deforms the drops. \nA recent study measured the deformation of two nearby drops with an applied DC electric potential difference \\cite{Bird2009}. \nTheir experiments showed that the drops sharpen into cones, and they measured a cone angle of roughly $20^{\\circ}$ for a potential difference of $500$ V (where $0^{\\circ}$ corresponds to no deformation). \nTheir measurements of the cone angle are approximately linear for electric potentials between $0$ and $500$ V, suggesting that the cone angle would be less than $0.08^{\\circ}$ for the applied voltages used in this work ($V_{\\text{in}}\\leq 2$ V). \n(The angle is likely diminished even further since the measurements in this work are AC instead of DC.) \nThus, any deformation of the drops is expected to be on a small scale, although it could contribute to forming the initial microscopic neck for $V_{\\text{in}}>0.3$ V. \n\nFigure \\ref{CfinalVa}(b) shows that at lower voltages, $V_{\\text{in}}<0.3$ V, $C_{\\text{final}}$ is roughly constant within error, and the description invoking a threshold electric field is a poor fit. \nInstead, the data are consistent with a picture where Van der Waals forces initiate coalescence at finite separation when $V_{\\text{in}}$ is small.\nFor the data at low voltages, I measure $C_{\\text{final}}=0.63 \\pm 0.05$ pF, giving a best fit of $z_0=280$ nm. \nBecause the capacitance is logarithmic in drop separation, the experimental error on $z_0$ is large; the data are consistent with $z_0$ ranging from $120$ nm to $650$ nm. \n\n\n\n\n\\subsection{Initial neck size}\n\nFinally, I address the finite length and width of the liquid neck that is formed at the inception of coalescence. \nDue to the finite separation of the drops at the moment of contact, the separation between the drop interfaces at radius $r$ will be given by $r^2\/A+z_0$, instead of simply $r^2\/A$ as was assumed in previous sections. \nHowever, this correction becomes relatively smaller as $r_{\\text{min}}$ grows. \nNumerical simulations \\cite{Baroudi2013} where low-viscosity drops initiate contact by forming a small fluid neck at finite separation show that after a short delay, the dynamics converge onto the predicted scaling (i.e. Eq.\\ (\\ref{invScaling})). \n\nPrevious high-speed imaging studies have reported values for the initial finite radius of the fluid neck (referred to as $r_0$) at the inception of liquid drop coalescence in air. \nValues reported were $r_0=50$ $\\mu$m for $U_{\\text{app}} \\lesssim 0.1$ mm\/s (ref.\\ \\cite{Thoroddsen2005}) and $r_0=43.8 \\pm 4.3$ $\\mu$m for $U_{\\text{app}}=6.6$ mm\/s (ref.\\ \\cite{Fezzaa2008}). \nIn contrast, I measure $r_{\\text{min}}$ down to $0.7$ $\\mu$m at $\\tau=50$ ns for aqueous NaCl drops at low approach-speed.\nThis is a significantly smaller upper bound for the initial size of the neck for aqueous NaCl drops at low approach-speed.\n\n\n\n\n\n\\section{Conclusion}\n\nIn summary, I have presented supporting evidence for the new phase diagram for liquid drop coalescence in vacuum or air, developed in refs.\\ \\cite{Paulsen2012, Paulsen2011}. \nThe theoretically unanticipated inertially limited viscous regime was found to be the true asymptotic regime of coalescence for drops of any finite viscosity. \nIn this regime, surface-tension, viscosity, and inertia all balance. \nViscous drops ($Oh>1$) transition into the Stokes regime once the neck is sufficiently large to pull the drops towards each other. \nLow-viscosity drops ($Oh<1$) transition into the inertial regime at late times. \n\nIn the inertially limited viscous regime and the Stokes regime, the center-of-mass motion of the drops was found to track with the motion of the backs of the drops, further solidifying the force balance argument that identified the inertially limited viscous regime in ref.\\ \\cite{Paulsen2012}. \nThis work provides similarity solutions for the neck shapes in these two regimes, and the new phase diagram for coalescence was shown to apply for different boundary conditions. \n\nAdditional evidence was provided for the surprisingly late viscous-to-inertial crossover (from the inertially limited viscous regime to the inertial regime), including an alternative method of data-collapse, a Reynolds-number argument coming from the viscous side, and high-speed imaging experiments where the surface tension was varied. \nThe agreement of the new coalescence Reynolds number with the data supports the new picture for the flows, which must have a significant gradient on a small axial length scale set by the neck height, $r_{\\text{min}}^2\/A$. \n\nMany of the results are based on electrical measurements, which were shown to have an insignificant effect on the coalescence dynamics reported here. \nAt low approach-speed and low applied-voltage, the drops coalesce at finite separation as undeformed spheres. \n\nWhereas this work has established the behavior of liquid drop coalescence in vacuum or air, further work is needed to determine how an outer fluid with significant density or viscosity alters the coalescence phase diagram. \n\n\n\n\n\n\n\\begin{acknowledgments} \nI thank Sidney Nagel and Justin Burton for their guidance, support, and keen insight throughout this work. \nI am also particularly grateful to Santosh Appathurai, Osman Basaran, Sarah Case, Thomas Rosenbaum, Savdeep Sethi, and Wendy Zhang. \nI thank Michelle Driscoll, Efi Efrati, Nathan Keim, and Tom Witten for their assistance and for many illuminating discussions. \nThis work was supported by NSF Grant DMR-1105145 and by NSF MRSEC DMR-0820054. \n\\end{acknowledgments}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Int}\n\nExoplanets -- planets orbiting stars other than the Sun -- are most often identified through indirect means. We observe a star with periodic behaviour that would otherwise be unexpected and conclude that the best explanation is the presence of one (or more) planets. We direct the interested reader to \\cite{2018Perryman} for a summary of the various exoplanet detection techniques. By piecing together the observations of the unexpected behaviour it is possible to constrain, to some degree, the orbit and physical nature of the planets in question. Such inference is, however, not perfect -- particularly when the planets in question have been detected through observations of the ``wobble'' of their host star, as is the case for planets found using the radial velocity technique \\citep[e.g. ][]{51peg,47UMa,HarpsM,114613,3167c}, or candidate planets claimed on the basis of binary star eclipse timing variability \\citep[e.g. ][]{hwvir,nnser,silly1,uzfor}.\n\nThe accurate determination of the (minimum) masses and orbits of newly discovered exoplanets provides the key data by which we can understand the variety of outcomes of the planet formation process. As such, it behooves us to ensure that exoplanet catalogues contain information which is as accurate and realistic as possible. Such accurate solutions do not just enable us to properly ascertain the distribution of planets at the current epoch -- they also provide an important window into the history of the planetary systems we discover \\citep[e.g. ][]{2014Ford,2015PuWu,2017Fulton,2019Wu}, and allow us to predict and plan follow up observations through population synthesis models \\citep[e.g. ][]{2013Hasegawa,2018Mordasini,2020Dulz}. For example, the migration and mutual gravitational interaction of planets have been identified as being of critical importance to both the observed architectures and predicted long-term stability of the menagerie of known multi-planet systems heretofore identified through radial velocity and transit surveys \\citep[e.g.][]{prototrap,2012cWittenmyer,chain,trappist,2017Mustill,2017Hamers,2019Childs}. \n\nHowever, the accuracy of orbital parameters of the planetary companions presented in discovery works is frequently limited by the time period covered by the observations that led to the discovery, which are often enough to claim detection and little more \\citep{JupiterAnalogues,CoolJupiters}. Long term follow-up of known planet host systems is therefore desirable to refine the orbital parameters for known companions, to infer the presence of additional companions at lower masses and\/or larger semi-major axes \\citep[e.g.][]{2017BeckerAdams,EightSingle,RVDItech,181433,2019Denham,Rick19}, and to disentangle the complex signals produced by planets on resonant \\citep[e.g.][]{Ang10,2012cWittenmyer,2014aWittenmyer,2016Wittenmyer} or eccentric orbits \\citep[e.g.][]{2013Wittenmyer,2017cWittenmyer}. Equally, due to the relative paucity of data on which planet discoveries are often based, it is possible for those initial solutions to change markedly as more data are acquired. The ultimate extension of this is that, on occasion, the process by which planetary solutions are fit to observational data can yield false solutions -- essentially finding local minima in the phase space of all possible orbital solutions that represent a good theoretical fit to the data whilst being unphysical. It is therefore important to check the dynamical feasibility of multi-planet solutions that appear to present a good fit to observational data -- particularly in those cases where such solutions invoke planets on orbits that offer the potential for close encounters between the candidate planets.\n\nFollowing this logic, we have in the past tested the stability of multi-planet systems in a variety of environments including around main sequence \\citep{2010Marshall,2012bWittenmyer,2015Wittenmyer,2017bWittenmyer}, evolved \\citep{2017aWittenmyer,2017cWittenmyer,2019Marshall}, and post-main sequence stars \\citep{2011Horner,2012Horner,QSVir,2012aWittenmyer,2013Mustill}. In some cases, our results confirmed that the proposed systems were dynamically feasible as presented in the discovery work, whilst in others, our analysis demonstrated that alternative explanations must be sought for the observed behaviour of the claimed ``planet-host'' star \\citep[e.g.][]{2011Horner,QSVir}. To ensure that our own work remains robust, we have incorporated such analysis as a standard part of our own exoplanet discovery papers. We test all published multi-planet solutions for dynamical stability before placing too great a confidence in a particular outcome. As an extension to this approach, we presented a revised Bayesian method to the previously adopted frequentist stability analysis in \\cite{2019Marshall}, and demonstrated the consistency between these approaches. \n\nRather than using direct dynamical simulations, the stability of a planetary system can also be inferred from a criterion derived from the planetary masses, semi-major axes, and conservation of the angular momentum deficit \\citep[AMD,][]{2000Laskar,2017Laskar}. AMD can be interpreted as measuring the degree of excitation of planetary orbits, with less excited orbits implying greater stability. The definition of AMD stability has been revised to account for the effect of mean motion resonances and close encounters on orbital stability \\citep{2017Petit,2018Petit}. Of the systems examined in this work, HD~110014 has been identified as being weakly stable, whilst HD~67087 and HD~133131A are both considered unstable according to AMD \\citep[see Figs. 6 and 7,][]{2017Laskar}. In our previous dynamical studies, we find good agreement between the stability inferred from AMD and our dynamical simulations with 13 systems in common between them, of which nine were classified unstable and four stable, one marginally so \\citep{2012Horner,2012Robertson,2012aWittenmyer,2012bWittenmyer,2012cWittenmyer,2014aWittenmyer,2014bWittenmyer,2015Wittenmyer,2016Wittenmyer,2016Endl}. In this paper we examine the dynamical stability of the three multi-planet systems, HD~67087, HD~110014, and HD~133131A, as a critical examination of their stability and a further test of the reliability of AMD for the identification of instability in exoplanet systems. \n\nHD~67087, observed as part of the Japanese Okayama Planet Search programme \\citep{2005Sato}, was discovered to host a pair of exoplanets by \\cite{2015Harakawa}. The candidate planets are super-Jupiters, with $m~\\sin i$ of 3.1\\,$M_{\\rm Jup}$\\ and 4.9\\,$M_{\\rm Jup}$, respectively.  They move on orbits with (${a,e}$) of ({1.08, 0.17}) and ({3.86, 0.76}), respectively, which would place the outer planet amongst the most eccentric Jovian planets identified thus far. The authors noted that the orbit and mass of the outer planet are poorly constrained.   \n\nHD\\,110014 was found to host a planet by \\citet{2009deMedeiros}; the second companion was identified through re-analysis of archival spectra taken by the FEROS instrument \\citep{1998KauferPasquini} looking to derive an updated orbit for planet b \\citep{2015Soto}.  The two candidate planets have super-Jupiter masses, and \\citet{2015Soto} cautioned that the proposed second planet was worryingly close in period to the typical rotation period of K giant stars. However, their analysis of the stellar photometry was inconclusive in identifying its activity as the root cause for the secondary signal. \n\nHD~133131A's planetary companions were reported in \\citet{2016Teske}, based on precise radial velocities primarily from the Magellan Planet Finder Spectrograph \\citep{2006Crane,2008Crane,2010Crane}. Their data supported the presence of two planets, where the outer planet is poorly constrained due to its long period. \\citet{2016Teske} ran a single dynamical stability simulation on the adopted solution and found it to remain stable for the full $10^5$ yr duration. The authors presented both a low- and high- eccentricity solution, reasoning that in a formal sense, the two solutions were essentially indistinguishable. They favoured the low-eccentricity model ($e_2=0.2$) for dynamical stability reasons. There is precedent in the literature for this choice, since it does happen that the formal best fit can be dynamically unfeasible whilst a slightly worse fit pushes the system into a region of stability \\citep[e.g.][]{2013Mustill, 2014Trifonov, 2017bWittenmyer}.  \n\nThe remainder of the paper is laid out as follows. We present a brief summary of the radial velocity observations and other data (e.g. stellar parameters) used for our reanalysis in Sect. \\ref{sec:Obs} along with an explanation of our modelling approach. The results of the reanalyses for each target are shown in Sect. \\ref{sec:Res}. A brief discussion of our findings in comparison to previous work on these systems is presented in Sect. \\ref{sec:Dis}. Finally, we present our conclusions in Sect. \\ref{sec:Con}.\n\n\\section{Observations and methods}\n\\label{sec:Obs}\n\n\\subsection{Radial Velocity Data}\n\nWe compiled radial velocity values from the literature for the three systems examined in this work; the origin of these data are summarised in Table \\ref{tab:rvs}. \n\n\\begin{table}\n\\centering\n\\caption{Table of references for the radial velocities data used in this work. \\label{tab:rvs}}\n\\begin{tabular}{ll}\n    \\hline\n    Target & References \\\\\n    \\hline\\hline\n    HD~67087   & \\cite{2015Harakawa} \\\\\n    HD~110114  & \\cite{2009deMedeiros} \\\\\n    HD~133131A & \\cite{2016Teske} \\\\\n    \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table*}\n\\centering\n\\caption{Planetary orbital parameters based on \\textit{Systemic} fits to radial velocity data. Semi-major axes were calculated using measured orbital periods and stellar masses taken from the NASA Exoplanet Archive. \\label{tab:systemic}}\n\\begin{tabular}{lcccccc}\n    \\hline\n    & HD\\,67087 b & HD\\,67087 c & HD\\,110014 b & HD\\,110014 c & HD\\,133131A b & HD\\,133131A c \\\\\n    \\hline \\hline\n    Amplitude [\\mbox{m\\,s$^{-1}$}] &  74.0~$\\pm$~3.0 & 54.0~$\\pm$~4.0 & 36.949$\\pm$0.750 & 5.956$\\pm$1.617 & 135.315$\\pm$3.640 & 64.660$\\pm$3.966 \\\\\n    Period [days] & 352.3$\\pm$1.7 & 2380$^{+167}_{-141}$ & 877.5$\\pm$5.2 & 130.125$\\pm$0.096 & 648.$\\pm$3 & 5342$^{+7783}_{-2009}$ \\\\\n    Mean anomaly [deg] & 35$^{+20}_{-16}$ & 94$^{+28}_{-35}$ & 155$\\pm$4 & 231$\\pm$3 & 265$\\pm$11 & 188$^{+104}_{-123}$ \\\\\n    Longitude [deg] & 281$^{+18}_{-15}$ & 256 (fixed) & 41$\\pm$3 & 302$\\pm$4 & 16.6$^{+4.7}_{-4.5}$ & 110$^{+25}_{-43}$ \\\\\n    Eccentricity & 0.18$^{+0.07}_{-0.06}$ & 0.51 (fixed) & 0.259$\\pm$0.017 & 0.410$\\pm$0.022 & 0.340$\\pm$0.032 & 0.63$^{+0.25}_{-0.20}$ \\\\\n    $M$ sin $i$ [$M_{\\rm Jup}$] & 3.10$^{+0.15}_{-0.14}$ & 3.73$^{+0.47}_{-0.45}$ & 10.61$\\pm$0.25 & 3.228$\\pm$0.098 & 1.418$\\pm$0.036 & 0.52$^{+0.45}_{-0.17}$ \\\\\n    Semi-major Axis [au] & 1.08 & 3.87 & 2.32 & 0.65 & 1.44 & 5.88 \\\\\n    \\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}\n\\centering\n\\caption{Results from \\textsc{Astroemperor} exploration of parameter space around \\textit{Systemic} nominal best fit values for planetary companions to HD~133131A and HD~110014. \\label{tab:mcmc}}\n\\begin{tabular}{lcccc}\n    \\hline\n    & HD\\,133131A b & HD\\,133131A c & HD\\,110014 b & HD\\,110014 c \\\\\n    \\hline \\hline\n    Amplitude [\\mbox{m\\,s$^{-1}$}] & 36.949$\\pm$0.750 & 5.956$\\pm$1.617 & 135.315$\\pm$3.640 & 64.660$\\pm$3.966 \\\\\n    Period [days] & 647.816$\\pm$1.575 & 3205.648$\\pm$948.063 & 865.206$\\pm$6.170 & 132.431$\\pm$0.279 \\\\\n    Phase [deg] & 261.620$\\pm$4.850 & 31.734$\\pm$98.433 & 43.753$\\pm$72.179 & 341.373$\\pm$64.346 \\\\\n    Longitude [deg] & 18.550$\\pm$2.165 & 113.777$\\pm$81.302 & 146.633$\\pm$17.229 & 236.903$\\pm$16.452 \\\\\n    Eccentricity & 0.341$\\pm$0.021 & 0.263$\\pm$0.145 & 0.011$\\pm$0.015 & 0.294$\\pm$0.076 \\\\\n    M sin $i$ [$M_{\\rm Jup}$]& 1.428$\\pm$0.099 & 0.388$\\pm$0.124 & 10.622$\\pm$0.757 & 2.581$\\pm$0.247 \\\\\n    Semimajor Axis [au] & 1.435$\\pm$0.046 & 4.153$\\pm$0.800 & 2.350$\\pm$0.075 & 0.668$\\pm$0.023 \\\\\n    \\hline\n    Jitter [\\mbox{m\\,s$^{-1}$}] & 3.557$\\pm$1.254 & 0.466$\\pm$0.419 & 6.060$\\pm$1.856 & 13.350$\\pm$1.492 \\\\\n    Offset [\\mbox{m\\,s$^{-1}$}] & -9.333$\\pm$4.787 & 12.321$\\pm$7.543 & 52.575$\\pm$4.737 & 72.198$\\pm$4.541 \\\\\n    MA coefficient & 0.714$\\pm$0.531 & 0.466$\\pm$0.419 & 0.697$\\pm$0.214 & 13.350$\\pm$1.492 \\\\\n    MA Timescale [days] & 4.158$\\pm$2.815 & 12.321$\\pm$7.543 & 9.793$\\pm$2.488 & 72.198$\\pm$4.541 \\\\\n    Acceleration [\\mbox{m\\,s$^{-1}$}\/yr] & -1.435 & & -21.620 & \\\\\n    \\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Modelling}\n\nTo test the dynamical stability of these proposed planetary systems, we follow the updated dynamical methodology outlined in our previous work \\citep[][]{2017bWittenmyer,2019Marshall}.\n\nIn brief, we perform a fit to the published velocity data using the \\textit{Systemic Console} \\citep{2009Meschiari}. We then use the MCMC tool within \\textit{Systemic} to explore the parameter space about the best fit. The MCMC chain runs for $10^7$ steps, discarding the first 10,000, and we then draw the trial solutions for our dynamical stability simulations from these posteriors. Using this data, we populate three ``annuli'' in $\\chi^2$ space corresponding to the ranges $0-1\\sigma$, $1-2\\sigma$, and $2-3\\sigma$ from the best fit.  Each annulus contains 42,025 unique realisations drawn from the MCMC chain. The innermost annulus was drawn from the lowest 68.3 per cent of all $\\chi^2$ values, the middle annulus contained the next best 27.2 per cent of values, and the outer annulus contained the worst 4.5 per cent of solutions (i.e. those falling $2-3\\sigma$ away from the best fit). The result is a set of ``clones'' which fall within $3\\sigma$ of the best-fit solution, thus representing a reasonable region of parameter space within which we explore the dynamical stability of the proposed planetary system, using the constraints afforded by the existing observational data.\n\nWe then proceed to perform lengthy dynamical simulations of each of the 126,075 solutions generated by this method. We used the Hybrid integrator within the $n$-body dynamics package \\textit{Mercury} \\citep{1999Chambers} to integrate the solutions forwards in time for a period of 100 Myr. The simulations are brought to a premature end if either of the planets being simulated is ejected from the system, is flung in to the central star, or if the two planets collide with one another. When such events occur, the time at which the collision or ejection occurred is recorded, giving us the lifetime for that particular run. As such, our suite of simulations yield 126,075 tests of the candidate planetary system, allowing us to study how its stability varies as a function of the particular details of the solution chosen to explain the observational data.\n\nWe determine the best-fit parameters and uncertainties for each system using the code Exoplanet Mcmc Parallel tEmpering Radial velOcity fitteR\\footnote{\\href{https:\/\/github.com\/ReddTea\/astroEMPEROR}{https:\/\/github.com\/ReddTea\/astroEMPEROR}} ({\\sc astroEMPEROR}), which uses thermodynamic methods combined with MCMC. Our approach has previously been established and described in \\cite{2019Marshall} and \\cite{EightSingle}. We summarise the input values and constraints used in the fitting presented in this work for the sake of reproducibility. Given that our goal was to test the feasibility of the exoplanetary systems as presented in the literature, we restricted {\\sc astroEMPEROR} to consider zero, one, or two planetary signals in the radial velocity data; dynamical configurations with additional planetary companions in orbits that could mimic a single planetary companion, e.g. two resonant planets looking like a single eccentric planet (for a total of three planetary companions), were not considered in this analysis. The planetary fitting parameters were the orbital period ($P$), line-of-sight mass ($M\\sin i$), orbital eccentricity ($e$), longitude of periastron ($\\omega$), and mean anomaly ($M$). We also include an additional jitter term when fitting the data. We initialised the locations of the walkers in the MCMC fitting at their best-fit values from the \\textit{Systemic} console fit, plus a small random scatter. The priors on each parameter were flat and unbounded i.e. with uniform probability between $\\pm\\infty$, except for the orbital eccentricities which had folded Gaussian priors, and the jitter term, which was a Jeffries function (but still unbound between $\\pm\\infty$). The parameter space was surveyed by 150 walkers at five temperatures over 15\\,000 steps, with the first 5\\,000 steps being discarded as the burn-in phase.\n\n\\section{Results}\n\\label{sec:Res}\n\n\\subsection{HD~67087}\n\nThe HD~67087 system is catastrophically unstable, as illustrated by the results of our stability analysis in Fig. \\ref{fig:HD67087_stability}. In this plot it is clear that the most stable solutions cluster toward the largest ratios of semi-major axes, and the smallest eccentricities. Even in this limit, the longest-lived solutions that plausibly represent the observations are still only stable for 10$^{6}$ yrs, out of a total integration time of 10$^{8}$ yrs. This leads us to the interpretation that the HD~67087 system, as inferred from the available radial velocity data, is dynamically infeasible. Given this high degree of instability, we do not attempt to determine a global best-fit solution for the system parameters.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD67087_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD67087_MRatio_MaxEcc_200_45.png}\n    \\caption{Visualisation of the dynamical stability of the HD~67087 planetary system. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~67087b and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~67087b and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. We find no stable solutions that last the full 100 Myr duration of the dynamical simulations close to the nominal best-fit orbital solution for the planets, with the only stable solutions lying at the extreme edges of the parameter space toward low eccentricities, large separations and low mass ratios. \\label{fig:HD67087_stability}}\n\\end{figure*} \n\n\\subsection{HD~110014}\n\nThe HD~110014 system is found to be dynamically stable, with a broad swathe of parameter space centred on the nominal solution producing system architectures that last for the full 10$^{8}$ yrs of our dynamical integrations. We show the results of the stability analysis, sampling the 3-$\\sigma$ parameter space around the nominal orbital solution determined from the radial velocities in Fig. \\ref{fig:HD110014_stability}. The results of the Bayesian analysis, showing what we infer to be the global best-fit parameters for the system, are presented in Fig. \\ref{fig:HD110014_confusogram}.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD110014_MaxEcc_ARatio_1800_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD110014_MRatio_MaxEcc_1800_45.png}\n    \\caption{Visualisation of the dynamical stability of the HD~110014 planetary system. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~110014b and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~110014b and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. We find stable solutions that last the full 100 Myr duration of the dynamical simulations close to the nominal best-fit orbital solution for the planets. \\label{fig:HD110014_stability}}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\t\\includegraphics[width=\\textwidth]{HD110014Triangle.pdf}\n\t\\caption{Bayesian posterior distributions of HD~110014 b and HD~110014 c's orbital parameters derived from {\\sc astroemperor}. From left to right (top to bottom), the parameters are $K_{b}$, $P_{b}$, $\\omega_{b}$, $\\phi_{b}$, $e_{b}$, $K_{c}$, $P_{c}$, $\\omega_{c}$, $\\phi_{c}$ and $e_{c}$. Credible intervals are denoted by the solid contours with increments of 1-$\\sigma$.}\n    \\label{fig:HD110014_confusogram}\n\\end{figure*}\n\n\\subsection{HD~133131A}\n\nThe HD~133131A system shows a very complex parameter space in the stability plots. As one would expect, the stability of the system generally increases towards lower orbital eccentricities and lower mass ratios between the two planetary components. The overall stability appears to be insensitive to the ratio of the semi-major axes for the planets, with long-lived solutions possible across the full range of values probed for this parameter. Interestingly, we demonstrate that stable architectures for the planetary system exist in both the high and low orbital eccentricity scenarios for the system. We show the results of the stability analysis, sampling the 3-$\\sigma$ parameter space around the nominal orbital solution determined from the radial velocities in Fig. \\ref{fig:HD133131A_stability}. The results of the Bayesian analysis, showing what we infer to be the global best-fit parameters for the system, are presented in Fig. \\ref{fig:HD133131A_confusogram}. Further observations to refine the planet properties of this system will be required to definitively characterise its dynamical stability.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD133131AH_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AH_MRatio_MaxEcc_200_45.png}\\\\\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AL_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AL_MRatio_MaxEcc_200_45.png}\n    \\caption{Plots of the dynamical stability of the HD~133131A planetary system for both the high eccentricity (top) and low eccentricity (bottom) orbital solutions. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~133131Ab and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~133131Ab and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. The stability revealed by our dynamical simulations is complex, with regions of both extreme stability (log(lifetime) $\\sim$ 100 Myrs) and instability (log(lifetime) $\\sim$ 100 yrs) lying within the 3-$\\sigma$ reach of the nominal best-fit orbital parameters.  \\label{fig:HD133131A_stability}}\n\\end{figure*} \n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{HD133131ATriangle.pdf}\n\t\\caption{Bayesian posterior distributions of HD~133131A b and HD~133131A c's orbital parameters derived from {\\sc astroemperor}. From left to right (top to bottom), the parameters are $K_{b}$, $P_{b}$, $\\omega_{b}$, $\\phi_{b}$, $e_{b}$, $K_{c}$, $P_{c}$, $\\omega_{c}$, $\\phi_{c}$ and $e_{c}$. Credible intervals are denoted by the solid contours with increments of 1-$\\sigma$.}\n    \\label{fig:HD133131A_confusogram}\n\\end{figure*}\n\n\\section{Discussion}\n\\label{sec:Dis}\n\nThe results of our dynamical modelling for the three systems considered in this work, HD~67087, HD~110014 and HD~133131A show three distinctly different outcomes. For the first system tested, HD~67087, we find no orbital solutions that exhibit long-term dynamical stability. As a result, we are forced to conclude that, if the planets proposed to orbit that star are real, they must move on orbits significantly different from those proposed in the discovery work, and sampled in our simulations. It seems likely that new radial velocity observations of HD~67087, extending the temporal baseline over which the star has been observed, will yield fresh insights to the system -- either significantly constraining and altering the proposed orbit for the outermost planet, or even revealing that that eccentric solution is in fact the result of multiple unresolved planets at large orbital radii. Such an outcome is far from unusual -- and indeed, it is often the case that, with more data, a single eccentric planet seen in RV data is resolved to actually be two planets moving on near circular orbits \\citep[e.g.][]{2013Wittenmyer,EightSingle}. For now, however, we can do no more than to call the existence of HD~67087~c into question, pending the acquisition of such additional data.\n\nIn contrast to the instability of HD~67087, our simulations of the HD~110014 system reveal that the best-fit solution for that two-planet system lies in a broad region of strong dynamical stability. In this case, our simulations simply reveal that the system, as proposed in the discovery work, is dynamically feasible -- and in a sense, the simulations add little beyond that.\n\nThe case of HD~133131A is somewhat more interesting. Here, our simulations reveal that solutions that fit the observational data can exhibit both strong dynamical stability, and extreme instability (with dynamical lifetimes of just a few years). Both the high- and low-eccentricity solutions considered in \\citet{2016Teske} can produce scenarios that are stable for the full 100 Myr of our simulations. In both the high- and low-eccentricity cases, the stable solutions cluster around the least eccentric available scenarios. The more widely separated the two planets, the more eccentric their orbits can be before instability occurs -- a natural result of the stability being driven by the minimum separation between the planets, rather than their orbital semi-major axes. The more widely the semi-major axes of the orbits are spaced, the more eccentric they must be to bring the planets into close proximity. These results show once again the benefits inherent to such dynamical analysis -- reminding us how studying the dynamical evolution of a given system can help to provide stronger constraints on the orbits of the planets contained therein than is possible by studying the observational data on their own.\n\nA comparison of our results to the analysis of the AMD stability criterion presented in \\cite{2017Laskar} shows agreement between the two different techniques for the dynamical stability of the three systems. Whilst HD~67087 and HD~110014 are respectively very clear cut cases of an unstable and a stable system, HD~133131A exhibits a more complex behaviour. HD~133131A may be dynamically stable, but the inferred lifetime for the planetary system as proposed is sensitive to the chosen initial conditions; this system therefore represents an edge case of stability where limitations of available data and the respective analyses provide no clear answer to the veracity of the previously inferred planetary system.\n\nCombining these new results with our previous dynamical analyses, as summarised in the introduction, we may consider that the AMD criterion is a reliable estimator of stability for planetary systems. There are 13 systems (out of 131 considered in that work) from \\cite{2017Laskar} that have had dynamical modelling of their stability. In \\cite{2017Laskar}, a planetary system is considered strongly stable if all planet-pairs have $\\beta$ values less than 1, such that collisions are impossible whilst weakly stable planetary systems are those in which the inner-most planet might collide with the star without disrupting the remainder of the planetary system. In five systems, both the AMD criterion and dynamical modelling agree on their dynamical stability (HD~142, HD~159868, NN Ser (AB), GJ~832, and HD~110014); the planets in each of these systems are dynamically well separated and therefore not strongly interacting \\citep[][this work]{2011Horner,2012bWittenmyer,2014bWittenmyer}. Six systems are unstable according to the AMD criterion with values of $\\beta$ in the range 1 to 5 for the planet pair (HD~155358, 24 Sex, HD~200964, HD~73526, HD~33844, HD~47366), but all are in mean motion resonances and have been demonstrated to be dynamically stable through $n$-body simulations \\citep{2012Robertson,2012cWittenmyer,2014aWittenmyer,2016Wittenmyer,2019Marshall}. The remaining two systems (HD~67087, HD~133131A) are dynamically unstable in both the AMD and dynamical analysis (this work). However, dynamical analysis of the HD~133131A system reveals regions of dynamical stability consistent with the observed radial velocities, prompting the need for further investigation of this system and its architecture. Neither of these two unstable planetary systems have $\\beta$ values radically different from those of the planetary systems in resonance, or each other, such that determining their stability can only be carried out using dynamical simulations. The existence of such systems in the known planet population as demonstrated in our analysis therefore showcases the necessity of performing long duration dynamical analyses of proposed planetary system architectures to reveal the complex dynamical interplay between high mass planets, the evolution of their orbital elements, and determine what constraints this places on the available parameter space for the endurance of the proposed planetary system over its lifetime.\n\n\\section{Conclusions}\n\\label{sec:Con}\n\nWe re-analysed the dynamical stability of the exoplanet systems around HD~67087, HD~110014, and HD~133131A, using available radial velocity data. These three planetary systems have poorly constrained orbital parameters, and had previously been identified as being potentially unstable. We combine a determination of the best-fit orbital parameters from least-squares fitting to the data with $n$-body simulations to determine the global best-fit solution for the planetary system architectures, and thereafter determine the probability distribution of the orbital solutions through Bayesian inference. \n\nOur dynamical analysis confirms that the published planetary system parameters for HD~67087bc are dynamically unstable on very short timescales, and we must conclude that the system, as published, is dynamically unfeasible. As more data are collected for the HD~67087 system, it seems likely that the true nature of the candidate planets therein will be revealed, and that future planetary solutions for that system will veer towards dynamical stability as the planetary orbits become better constrained. \n\nIn the case of HD~110014 bc we demonstrate that the system parameters can be dynamically stable for the full duration of our 100 Myr integrations. The third system, HD~133131A , exhibits much more complex behaviour, with HD~133131A bc being strongly unstable over much of the parameter space exhibited in this work including the region encompassing the nominal best-fit to the orbital parameters. In agreement with previous analysis of this system, we strongly disfavour a high eccentricity orbital solution for planet c. Additional observations of this system will be required to more precisely determine the planetary properties for HD~133131A bc and thereby categorically rule on the plausibility of the proposed planetary system.\n\nThese results demonstrate the complementarity of various techniques to deduce the stability of planetary systems, with good agreement between the results of our various works, and that of the AMD approach. We highlight the appropriateness of dynamical simulations for determination the long-term stability of planetary systems in the presence of strongly interacting planets, which although costly in a computing sense capture the full essence of planetary interaction in such systems which is not possible with other techniques. We finally assert that the orbital parameters for these three systems which have been determined in this work (as summarised in Table 3) should be the accepted values adopted by exoplanet archives or elsewhere. This work is thus one additional thread in the tapestry of cross-checking of published results through various means that ensures the reliability of archival information on planetary properties and the architectures of planetary systems which are essential to inform models of the formation and evolution of the exoplanet population \\citep[e.g. ][]{2019Childs,2019Denham,2020He,2020VolkMalhotra}.\n\n\\section*{Acknowledgements}\n\nThis is a pre-copyedited, author-produced PDF of an article accepted for publication in MNRAS  following peer review. The version of record Marshall et al., 2020, MNRAS, 494, 2, 2280--2288 is available online \\href{https:\/\/academic.oup.com\/mnras\/article\/494\/2\/2280\/5819459}{here}.\n\nWe thank the anonymous referee for their comments which helped to improve the article.\n\nThis research has made use of NASA's Astrophysics Data System and the SIMBAD database, operated at CDS, Strasbourg, France.\n\nJPM acknowledges research support by the Ministry of Science and Technology of Taiwan under grants MOST104-2628-M-001-004-MY3 and MOST107-2119-M-001-031-MY3, and Academia Sinica under grant AS-IA-106-M03.\n\n\\textit{Software}: This research has made use of the following Python packages: \\textsc{matplotlib} \\citep{2007Hunter}; \\textsc{numpy} \\citep{2006Oliphant}; \\textsc{pygtc} \\citep{2016Bocquet}; \\textsc{emcee} \\citep{2013ForemanMackey}; \\textsc{corner} \\citep{2016ForemanMackey}; \\textsc{mercury} \\citep{1999Chambers}.\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\\label{sec:Int}\n\nExoplanets -- planets orbiting stars other than the Sun -- are most often identified through indirect means. We observe a star with periodic behaviour that would otherwise be unexpected and conclude that the best explanation is the presence of one (or more) planets. We direct the interested reader to \\cite{2018Perryman} for a summary of the various exoplanet detection techniques. By piecing together the observations of the unexpected behaviour it is possible to constrain, to some degree, the orbit and physical nature of the planets in question. Such inference is, however, not perfect -- particularly when the planets in question have been detected through observations of the ``wobble'' of their host star, as is the case for planets found using the radial velocity technique \\citep[e.g. ][]{51peg,47UMa,HarpsM,114613,3167c}, or candidate planets claimed on the basis of binary star eclipse timing variability \\citep[e.g. ][]{hwvir,nnser,silly1,uzfor}.\n\nThe accurate determination of the (minimum) masses and orbits of newly discovered exoplanets provides the key data by which we can understand the variety of outcomes of the planet formation process. As such, it behooves us to ensure that exoplanet catalogues contain information which is as accurate and realistic as possible. Such accurate solutions do not just enable us to properly ascertain the distribution of planets at the current epoch -- they also provide an important window into the history of the planetary systems we discover \\citep[e.g. ][]{2014Ford,2015PuWu,2017Fulton,2019Wu}, and allow us to predict and plan follow up observations through population synthesis models \\citep[e.g. ][]{2013Hasegawa,2018Mordasini,2020Dulz}. For example, the migration and mutual gravitational interaction of planets have been identified as being of critical importance to both the observed architectures and predicted long-term stability of the menagerie of known multi-planet systems heretofore identified through radial velocity and transit surveys \\citep[e.g.][]{prototrap,2012cWittenmyer,chain,trappist,2017Mustill,2017Hamers,2019Childs}. \n\nHowever, the accuracy of orbital parameters of the planetary companions presented in discovery works is frequently limited by the time period covered by the observations that led to the discovery, which are often enough to claim detection and little more \\citep{JupiterAnalogues,CoolJupiters}. Long term follow-up of known planet host systems is therefore desirable to refine the orbital parameters for known companions, to infer the presence of additional companions at lower masses and\/or larger semi-major axes \\citep[e.g.][]{2017BeckerAdams,EightSingle,RVDItech,181433,2019Denham,Rick19}, and to disentangle the complex signals produced by planets on resonant \\citep[e.g.][]{Ang10,2012cWittenmyer,2014aWittenmyer,2016Wittenmyer} or eccentric orbits \\citep[e.g.][]{2013Wittenmyer,2017cWittenmyer}. Equally, due to the relative paucity of data on which planet discoveries are often based, it is possible for those initial solutions to change markedly as more data are acquired. The ultimate extension of this is that, on occasion, the process by which planetary solutions are fit to observational data can yield false solutions -- essentially finding local minima in the phase space of all possible orbital solutions that represent a good theoretical fit to the data whilst being unphysical. It is therefore important to check the dynamical feasibility of multi-planet solutions that appear to present a good fit to observational data -- particularly in those cases where such solutions invoke planets on orbits that offer the potential for close encounters between the candidate planets.\n\nFollowing this logic, we have in the past tested the stability of multi-planet systems in a variety of environments including around main sequence \\citep{2010Marshall,2012bWittenmyer,2015Wittenmyer,2017bWittenmyer}, evolved \\citep{2017aWittenmyer,2017cWittenmyer,2019Marshall}, and post-main sequence stars \\citep{2011Horner,2012Horner,QSVir,2012aWittenmyer,2013Mustill}. In some cases, our results confirmed that the proposed systems were dynamically feasible as presented in the discovery work, whilst in others, our analysis demonstrated that alternative explanations must be sought for the observed behaviour of the claimed ``planet-host'' star \\citep[e.g.][]{2011Horner,QSVir}. To ensure that our own work remains robust, we have incorporated such analysis as a standard part of our own exoplanet discovery papers. We test all published multi-planet solutions for dynamical stability before placing too great a confidence in a particular outcome. As an extension to this approach, we presented a revised Bayesian method to the previously adopted frequentist stability analysis in \\cite{2019Marshall}, and demonstrated the consistency between these approaches. \n\nRather than using direct dynamical simulations, the stability of a planetary system can also be inferred from a criterion derived from the planetary masses, semi-major axes, and conservation of the angular momentum deficit \\citep[AMD,][]{2000Laskar,2017Laskar}. AMD can be interpreted as measuring the degree of excitation of planetary orbits, with less excited orbits implying greater stability. The definition of AMD stability has been revised to account for the effect of mean motion resonances and close encounters on orbital stability \\citep{2017Petit,2018Petit}. Of the systems examined in this work, HD~110014 has been identified as being weakly stable, whilst HD~67087 and HD~133131A are both considered unstable according to AMD \\citep[see Figs. 6 and 7,][]{2017Laskar}. In our previous dynamical studies, we find good agreement between the stability inferred from AMD and our dynamical simulations with 13 systems in common between them, of which nine were classified unstable and four stable, one marginally so \\citep{2012Horner,2012Robertson,2012aWittenmyer,2012bWittenmyer,2012cWittenmyer,2014aWittenmyer,2014bWittenmyer,2015Wittenmyer,2016Wittenmyer,2016Endl}. In this paper we examine the dynamical stability of the three multi-planet systems, HD~67087, HD~110014, and HD~133131A, as a critical examination of their stability and a further test of the reliability of AMD for the identification of instability in exoplanet systems. \n\nHD~67087, observed as part of the Japanese Okayama Planet Search programme \\citep{2005Sato}, was discovered to host a pair of exoplanets by \\cite{2015Harakawa}. The candidate planets are super-Jupiters, with $m~\\sin i$ of 3.1\\,$M_{\\rm Jup}$\\ and 4.9\\,$M_{\\rm Jup}$, respectively.  They move on orbits with (${a,e}$) of ({1.08, 0.17}) and ({3.86, 0.76}), respectively, which would place the outer planet amongst the most eccentric Jovian planets identified thus far. The authors noted that the orbit and mass of the outer planet are poorly constrained.   \n\nHD\\,110014 was found to host a planet by \\citet{2009deMedeiros}; the second companion was identified through re-analysis of archival spectra taken by the FEROS instrument \\citep{1998KauferPasquini} looking to derive an updated orbit for planet b \\citep{2015Soto}.  The two candidate planets have super-Jupiter masses, and \\citet{2015Soto} cautioned that the proposed second planet was worryingly close in period to the typical rotation period of K giant stars. However, their analysis of the stellar photometry was inconclusive in identifying its activity as the root cause for the secondary signal. \n\nHD~133131A's planetary companions were reported in \\citet{2016Teske}, based on precise radial velocities primarily from the Magellan Planet Finder Spectrograph \\citep{2006Crane,2008Crane,2010Crane}. Their data supported the presence of two planets, where the outer planet is poorly constrained due to its long period. \\citet{2016Teske} ran a single dynamical stability simulation on the adopted solution and found it to remain stable for the full $10^5$ yr duration. The authors presented both a low- and high- eccentricity solution, reasoning that in a formal sense, the two solutions were essentially indistinguishable. They favoured the low-eccentricity model ($e_2=0.2$) for dynamical stability reasons. There is precedent in the literature for this choice, since it does happen that the formal best fit can be dynamically unfeasible whilst a slightly worse fit pushes the system into a region of stability \\citep[e.g.][]{2013Mustill, 2014Trifonov, 2017bWittenmyer}.  \n\nThe remainder of the paper is laid out as follows. We present a brief summary of the radial velocity observations and other data (e.g. stellar parameters) used for our reanalysis in Sect. \\ref{sec:Obs} along with an explanation of our modelling approach. The results of the reanalyses for each target are shown in Sect. \\ref{sec:Res}. A brief discussion of our findings in comparison to previous work on these systems is presented in Sect. \\ref{sec:Dis}. Finally, we present our conclusions in Sect. \\ref{sec:Con}.\n\n\\section{Observations and methods}\n\\label{sec:Obs}\n\n\\subsection{Radial Velocity Data}\n\nWe compiled radial velocity values from the literature for the three systems examined in this work; the origin of these data are summarised in Table \\ref{tab:rvs}. \n\n\\begin{table}\n\\centering\n\\caption{Table of references for the radial velocities data used in this work. \\label{tab:rvs}}\n\\begin{tabular}{ll}\n    \\hline\n    Target & References \\\\\n    \\hline\\hline\n    HD~67087   & \\cite{2015Harakawa} \\\\\n    HD~110114  & \\cite{2009deMedeiros} \\\\\n    HD~133131A & \\cite{2016Teske} \\\\\n    \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table*}\n\\centering\n\\caption{Planetary orbital parameters based on \\textit{Systemic} fits to radial velocity data. Semi-major axes were calculated using measured orbital periods and stellar masses taken from the NASA Exoplanet Archive. \\label{tab:systemic}}\n\\begin{tabular}{lcccccc}\n    \\hline\n    & HD\\,67087 b & HD\\,67087 c & HD\\,110014 b & HD\\,110014 c & HD\\,133131A b & HD\\,133131A c \\\\\n    \\hline \\hline\n    Amplitude [\\mbox{m\\,s$^{-1}$}] &  74.0~$\\pm$~3.0 & 54.0~$\\pm$~4.0 & 36.949$\\pm$0.750 & 5.956$\\pm$1.617 & 135.315$\\pm$3.640 & 64.660$\\pm$3.966 \\\\\n    Period [days] & 352.3$\\pm$1.7 & 2380$^{+167}_{-141}$ & 877.5$\\pm$5.2 & 130.125$\\pm$0.096 & 648.$\\pm$3 & 5342$^{+7783}_{-2009}$ \\\\\n    Mean anomaly [deg] & 35$^{+20}_{-16}$ & 94$^{+28}_{-35}$ & 155$\\pm$4 & 231$\\pm$3 & 265$\\pm$11 & 188$^{+104}_{-123}$ \\\\\n    Longitude [deg] & 281$^{+18}_{-15}$ & 256 (fixed) & 41$\\pm$3 & 302$\\pm$4 & 16.6$^{+4.7}_{-4.5}$ & 110$^{+25}_{-43}$ \\\\\n    Eccentricity & 0.18$^{+0.07}_{-0.06}$ & 0.51 (fixed) & 0.259$\\pm$0.017 & 0.410$\\pm$0.022 & 0.340$\\pm$0.032 & 0.63$^{+0.25}_{-0.20}$ \\\\\n    $M$ sin $i$ [$M_{\\rm Jup}$] & 3.10$^{+0.15}_{-0.14}$ & 3.73$^{+0.47}_{-0.45}$ & 10.61$\\pm$0.25 & 3.228$\\pm$0.098 & 1.418$\\pm$0.036 & 0.52$^{+0.45}_{-0.17}$ \\\\\n    Semi-major Axis [au] & 1.08 & 3.87 & 2.32 & 0.65 & 1.44 & 5.88 \\\\\n    \\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}\n\\centering\n\\caption{Results from \\textsc{Astroemperor} exploration of parameter space around \\textit{Systemic} nominal best fit values for planetary companions to HD~133131A and HD~110014. \\label{tab:mcmc}}\n\\begin{tabular}{lcccc}\n    \\hline\n    & HD\\,133131A b & HD\\,133131A c & HD\\,110014 b & HD\\,110014 c \\\\\n    \\hline \\hline\n    Amplitude [\\mbox{m\\,s$^{-1}$}] & 36.949$\\pm$0.750 & 5.956$\\pm$1.617 & 135.315$\\pm$3.640 & 64.660$\\pm$3.966 \\\\\n    Period [days] & 647.816$\\pm$1.575 & 3205.648$\\pm$948.063 & 865.206$\\pm$6.170 & 132.431$\\pm$0.279 \\\\\n    Phase [deg] & 261.620$\\pm$4.850 & 31.734$\\pm$98.433 & 43.753$\\pm$72.179 & 341.373$\\pm$64.346 \\\\\n    Longitude [deg] & 18.550$\\pm$2.165 & 113.777$\\pm$81.302 & 146.633$\\pm$17.229 & 236.903$\\pm$16.452 \\\\\n    Eccentricity & 0.341$\\pm$0.021 & 0.263$\\pm$0.145 & 0.011$\\pm$0.015 & 0.294$\\pm$0.076 \\\\\n    M sin $i$ [$M_{\\rm Jup}$]& 1.428$\\pm$0.099 & 0.388$\\pm$0.124 & 10.622$\\pm$0.757 & 2.581$\\pm$0.247 \\\\\n    Semimajor Axis [au] & 1.435$\\pm$0.046 & 4.153$\\pm$0.800 & 2.350$\\pm$0.075 & 0.668$\\pm$0.023 \\\\\n    \\hline\n    Jitter [\\mbox{m\\,s$^{-1}$}] & 3.557$\\pm$1.254 & 0.466$\\pm$0.419 & 6.060$\\pm$1.856 & 13.350$\\pm$1.492 \\\\\n    Offset [\\mbox{m\\,s$^{-1}$}] & -9.333$\\pm$4.787 & 12.321$\\pm$7.543 & 52.575$\\pm$4.737 & 72.198$\\pm$4.541 \\\\\n    MA coefficient & 0.714$\\pm$0.531 & 0.466$\\pm$0.419 & 0.697$\\pm$0.214 & 13.350$\\pm$1.492 \\\\\n    MA Timescale [days] & 4.158$\\pm$2.815 & 12.321$\\pm$7.543 & 9.793$\\pm$2.488 & 72.198$\\pm$4.541 \\\\\n    Acceleration [\\mbox{m\\,s$^{-1}$}\/yr] & -1.435 & & -21.620 & \\\\\n    \\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Modelling}\n\nTo test the dynamical stability of these proposed planetary systems, we follow the updated dynamical methodology outlined in our previous work \\citep[][]{2017bWittenmyer,2019Marshall}.\n\nIn brief, we perform a fit to the published velocity data using the \\textit{Systemic Console} \\citep{2009Meschiari}. We then use the MCMC tool within \\textit{Systemic} to explore the parameter space about the best fit. The MCMC chain runs for $10^7$ steps, discarding the first 10,000, and we then draw the trial solutions for our dynamical stability simulations from these posteriors. Using this data, we populate three ``annuli'' in $\\chi^2$ space corresponding to the ranges $0-1\\sigma$, $1-2\\sigma$, and $2-3\\sigma$ from the best fit.  Each annulus contains 42,025 unique realisations drawn from the MCMC chain. The innermost annulus was drawn from the lowest 68.3 per cent of all $\\chi^2$ values, the middle annulus contained the next best 27.2 per cent of values, and the outer annulus contained the worst 4.5 per cent of solutions (i.e. those falling $2-3\\sigma$ away from the best fit). The result is a set of ``clones'' which fall within $3\\sigma$ of the best-fit solution, thus representing a reasonable region of parameter space within which we explore the dynamical stability of the proposed planetary system, using the constraints afforded by the existing observational data.\n\nWe then proceed to perform lengthy dynamical simulations of each of the 126,075 solutions generated by this method. We used the Hybrid integrator within the $n$-body dynamics package \\textit{Mercury} \\citep{1999Chambers} to integrate the solutions forwards in time for a period of 100 Myr. The simulations are brought to a premature end if either of the planets being simulated is ejected from the system, is flung in to the central star, or if the two planets collide with one another. When such events occur, the time at which the collision or ejection occurred is recorded, giving us the lifetime for that particular run. As such, our suite of simulations yield 126,075 tests of the candidate planetary system, allowing us to study how its stability varies as a function of the particular details of the solution chosen to explain the observational data.\n\nWe determine the best-fit parameters and uncertainties for each system using the code Exoplanet Mcmc Parallel tEmpering Radial velOcity fitteR\\footnote{\\href{https:\/\/github.com\/ReddTea\/astroEMPEROR}{https:\/\/github.com\/ReddTea\/astroEMPEROR}} ({\\sc astroEMPEROR}), which uses thermodynamic methods combined with MCMC. Our approach has previously been established and described in \\cite{2019Marshall} and \\cite{EightSingle}. We summarise the input values and constraints used in the fitting presented in this work for the sake of reproducibility. Given that our goal was to test the feasibility of the exoplanetary systems as presented in the literature, we restricted {\\sc astroEMPEROR} to consider zero, one, or two planetary signals in the radial velocity data; dynamical configurations with additional planetary companions in orbits that could mimic a single planetary companion, e.g. two resonant planets looking like a single eccentric planet (for a total of three planetary companions), were not considered in this analysis. The planetary fitting parameters were the orbital period ($P$), line-of-sight mass ($M\\sin i$), orbital eccentricity ($e$), longitude of periastron ($\\omega$), and mean anomaly ($M$). We also include an additional jitter term when fitting the data. We initialised the locations of the walkers in the MCMC fitting at their best-fit values from the \\textit{Systemic} console fit, plus a small random scatter. The priors on each parameter were flat and unbounded i.e. with uniform probability between $\\pm\\infty$, except for the orbital eccentricities which had folded Gaussian priors, and the jitter term, which was a Jeffries function (but still unbound between $\\pm\\infty$). The parameter space was surveyed by 150 walkers at five temperatures over 15\\,000 steps, with the first 5\\,000 steps being discarded as the burn-in phase.\n\n\\section{Results}\n\\label{sec:Res}\n\n\\subsection{HD~67087}\n\nThe HD~67087 system is catastrophically unstable, as illustrated by the results of our stability analysis in Fig. \\ref{fig:HD67087_stability}. In this plot it is clear that the most stable solutions cluster toward the largest ratios of semi-major axes, and the smallest eccentricities. Even in this limit, the longest-lived solutions that plausibly represent the observations are still only stable for 10$^{6}$ yrs, out of a total integration time of 10$^{8}$ yrs. This leads us to the interpretation that the HD~67087 system, as inferred from the available radial velocity data, is dynamically infeasible. Given this high degree of instability, we do not attempt to determine a global best-fit solution for the system parameters.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD67087_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD67087_MRatio_MaxEcc_200_45.png}\n    \\caption{Visualisation of the dynamical stability of the HD~67087 planetary system. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~67087b and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~67087b and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. We find no stable solutions that last the full 100 Myr duration of the dynamical simulations close to the nominal best-fit orbital solution for the planets, with the only stable solutions lying at the extreme edges of the parameter space toward low eccentricities, large separations and low mass ratios. \\label{fig:HD67087_stability}}\n\\end{figure*} \n\n\\subsection{HD~110014}\n\nThe HD~110014 system is found to be dynamically stable, with a broad swathe of parameter space centred on the nominal solution producing system architectures that last for the full 10$^{8}$ yrs of our dynamical integrations. We show the results of the stability analysis, sampling the 3-$\\sigma$ parameter space around the nominal orbital solution determined from the radial velocities in Fig. \\ref{fig:HD110014_stability}. The results of the Bayesian analysis, showing what we infer to be the global best-fit parameters for the system, are presented in Fig. \\ref{fig:HD110014_confusogram}.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD110014_MaxEcc_ARatio_1800_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD110014_MRatio_MaxEcc_1800_45.png}\n    \\caption{Visualisation of the dynamical stability of the HD~110014 planetary system. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~110014b and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~110014b and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. We find stable solutions that last the full 100 Myr duration of the dynamical simulations close to the nominal best-fit orbital solution for the planets. \\label{fig:HD110014_stability}}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\t\\includegraphics[width=\\textwidth]{HD110014Triangle.pdf}\n\t\\caption{Bayesian posterior distributions of HD~110014 b and HD~110014 c's orbital parameters derived from {\\sc astroemperor}. From left to right (top to bottom), the parameters are $K_{b}$, $P_{b}$, $\\omega_{b}$, $\\phi_{b}$, $e_{b}$, $K_{c}$, $P_{c}$, $\\omega_{c}$, $\\phi_{c}$ and $e_{c}$. Credible intervals are denoted by the solid contours with increments of 1-$\\sigma$.}\n    \\label{fig:HD110014_confusogram}\n\\end{figure*}\n\n\\subsection{HD~133131A}\n\nThe HD~133131A system shows a very complex parameter space in the stability plots. As one would expect, the stability of the system generally increases towards lower orbital eccentricities and lower mass ratios between the two planetary components. The overall stability appears to be insensitive to the ratio of the semi-major axes for the planets, with long-lived solutions possible across the full range of values probed for this parameter. Interestingly, we demonstrate that stable architectures for the planetary system exist in both the high and low orbital eccentricity scenarios for the system. We show the results of the stability analysis, sampling the 3-$\\sigma$ parameter space around the nominal orbital solution determined from the radial velocities in Fig. \\ref{fig:HD133131A_stability}. The results of the Bayesian analysis, showing what we infer to be the global best-fit parameters for the system, are presented in Fig. \\ref{fig:HD133131A_confusogram}. Further observations to refine the planet properties of this system will be required to definitively characterise its dynamical stability.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD133131AH_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AH_MRatio_MaxEcc_200_45.png}\\\\\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AL_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AL_MRatio_MaxEcc_200_45.png}\n    \\caption{Plots of the dynamical stability of the HD~133131A planetary system for both the high eccentricity (top) and low eccentricity (bottom) orbital solutions. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~133131Ab and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~133131Ab and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. The stability revealed by our dynamical simulations is complex, with regions of both extreme stability (log(lifetime) $\\sim$ 100 Myrs) and instability (log(lifetime) $\\sim$ 100 yrs) lying within the 3-$\\sigma$ reach of the nominal best-fit orbital parameters.  \\label{fig:HD133131A_stability}}\n\\end{figure*} \n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{HD133131ATriangle.pdf}\n\t\\caption{Bayesian posterior distributions of HD~133131A b and HD~133131A c's orbital parameters derived from {\\sc astroemperor}. From left to right (top to bottom), the parameters are $K_{b}$, $P_{b}$, $\\omega_{b}$, $\\phi_{b}$, $e_{b}$, $K_{c}$, $P_{c}$, $\\omega_{c}$, $\\phi_{c}$ and $e_{c}$. Credible intervals are denoted by the solid contours with increments of 1-$\\sigma$.}\n    \\label{fig:HD133131A_confusogram}\n\\end{figure*}\n\n\\section{Discussion}\n\\label{sec:Dis}\n\nThe results of our dynamical modelling for the three systems considered in this work, HD~67087, HD~110014 and HD~133131A show three distinctly different outcomes. For the first system tested, HD~67087, we find no orbital solutions that exhibit long-term dynamical stability. As a result, we are forced to conclude that, if the planets proposed to orbit that star are real, they must move on orbits significantly different from those proposed in the discovery work, and sampled in our simulations. It seems likely that new radial velocity observations of HD~67087, extending the temporal baseline over which the star has been observed, will yield fresh insights to the system -- either significantly constraining and altering the proposed orbit for the outermost planet, or even revealing that that eccentric solution is in fact the result of multiple unresolved planets at large orbital radii. Such an outcome is far from unusual -- and indeed, it is often the case that, with more data, a single eccentric planet seen in RV data is resolved to actually be two planets moving on near circular orbits \\citep[e.g.][]{2013Wittenmyer,EightSingle}. For now, however, we can do no more than to call the existence of HD~67087~c into question, pending the acquisition of such additional data.\n\nIn contrast to the instability of HD~67087, our simulations of the HD~110014 system reveal that the best-fit solution for that two-planet system lies in a broad region of strong dynamical stability. In this case, our simulations simply reveal that the system, as proposed in the discovery work, is dynamically feasible -- and in a sense, the simulations add little beyond that.\n\nThe case of HD~133131A is somewhat more interesting. Here, our simulations reveal that solutions that fit the observational data can exhibit both strong dynamical stability, and extreme instability (with dynamical lifetimes of just a few years). Both the high- and low-eccentricity solutions considered in \\citet{2016Teske} can produce scenarios that are stable for the full 100 Myr of our simulations. In both the high- and low-eccentricity cases, the stable solutions cluster around the least eccentric available scenarios. The more widely separated the two planets, the more eccentric their orbits can be before instability occurs -- a natural result of the stability being driven by the minimum separation between the planets, rather than their orbital semi-major axes. The more widely the semi-major axes of the orbits are spaced, the more eccentric they must be to bring the planets into close proximity. These results show once again the benefits inherent to such dynamical analysis -- reminding us how studying the dynamical evolution of a given system can help to provide stronger constraints on the orbits of the planets contained therein than is possible by studying the observational data on their own.\n\nA comparison of our results to the analysis of the AMD stability criterion presented in \\cite{2017Laskar} shows agreement between the two different techniques for the dynamical stability of the three systems. Whilst HD~67087 and HD~110014 are respectively very clear cut cases of an unstable and a stable system, HD~133131A exhibits a more complex behaviour. HD~133131A may be dynamically stable, but the inferred lifetime for the planetary system as proposed is sensitive to the chosen initial conditions; this system therefore represents an edge case of stability where limitations of available data and the respective analyses provide no clear answer to the veracity of the previously inferred planetary system.\n\nCombining these new results with our previous dynamical analyses, as summarised in the introduction, we may consider that the AMD criterion is a reliable estimator of stability for planetary systems. There are 13 systems (out of 131 considered in that work) from \\cite{2017Laskar} that have had dynamical modelling of their stability. In \\cite{2017Laskar}, a planetary system is considered strongly stable if all planet-pairs have $\\beta$ values less than 1, such that collisions are impossible whilst weakly stable planetary systems are those in which the inner-most planet might collide with the star without disrupting the remainder of the planetary system. In five systems, both the AMD criterion and dynamical modelling agree on their dynamical stability (HD~142, HD~159868, NN Ser (AB), GJ~832, and HD~110014); the planets in each of these systems are dynamically well separated and therefore not strongly interacting \\citep[][this work]{2011Horner,2012bWittenmyer,2014bWittenmyer}. Six systems are unstable according to the AMD criterion with values of $\\beta$ in the range 1 to 5 for the planet pair (HD~155358, 24 Sex, HD~200964, HD~73526, HD~33844, HD~47366), but all are in mean motion resonances and have been demonstrated to be dynamically stable through $n$-body simulations \\citep{2012Robertson,2012cWittenmyer,2014aWittenmyer,2016Wittenmyer,2019Marshall}. The remaining two systems (HD~67087, HD~133131A) are dynamically unstable in both the AMD and dynamical analysis (this work). However, dynamical analysis of the HD~133131A system reveals regions of dynamical stability consistent with the observed radial velocities, prompting the need for further investigation of this system and its architecture. Neither of these two unstable planetary systems have $\\beta$ values radically different from those of the planetary systems in resonance, or each other, such that determining their stability can only be carried out using dynamical simulations. The existence of such systems in the known planet population as demonstrated in our analysis therefore showcases the necessity of performing long duration dynamical analyses of proposed planetary system architectures to reveal the complex dynamical interplay between high mass planets, the evolution of their orbital elements, and determine what constraints this places on the available parameter space for the endurance of the proposed planetary system over its lifetime.\n\n\\section{Conclusions}\n\\label{sec:Con}\n\nWe re-analysed the dynamical stability of the exoplanet systems around HD~67087, HD~110014, and HD~133131A, using available radial velocity data. These three planetary systems have poorly constrained orbital parameters, and had previously been identified as being potentially unstable. We combine a determination of the best-fit orbital parameters from least-squares fitting to the data with $n$-body simulations to determine the global best-fit solution for the planetary system architectures, and thereafter determine the probability distribution of the orbital solutions through Bayesian inference. \n\nOur dynamical analysis confirms that the published planetary system parameters for HD~67087bc are dynamically unstable on very short timescales, and we must conclude that the system, as published, is dynamically unfeasible. As more data are collected for the HD~67087 system, it seems likely that the true nature of the candidate planets therein will be revealed, and that future planetary solutions for that system will veer towards dynamical stability as the planetary orbits become better constrained. \n\nIn the case of HD~110014 bc we demonstrate that the system parameters can be dynamically stable for the full duration of our 100 Myr integrations. The third system, HD~133131A , exhibits much more complex behaviour, with HD~133131A bc being strongly unstable over much of the parameter space exhibited in this work including the region encompassing the nominal best-fit to the orbital parameters. In agreement with previous analysis of this system, we strongly disfavour a high eccentricity orbital solution for planet c. Additional observations of this system will be required to more precisely determine the planetary properties for HD~133131A bc and thereby categorically rule on the plausibility of the proposed planetary system.\n\nThese results demonstrate the complementarity of various techniques to deduce the stability of planetary systems, with good agreement between the results of our various works, and that of the AMD approach. We highlight the appropriateness of dynamical simulations for determination the long-term stability of planetary systems in the presence of strongly interacting planets, which although costly in a computing sense capture the full essence of planetary interaction in such systems which is not possible with other techniques. We finally assert that the orbital parameters for these three systems which have been determined in this work (as summarised in Table 3) should be the accepted values adopted by exoplanet archives or elsewhere. This work is thus one additional thread in the tapestry of cross-checking of published results through various means that ensures the reliability of archival information on planetary properties and the architectures of planetary systems which are essential to inform models of the formation and evolution of the exoplanet population \\citep[e.g. ][]{2019Childs,2019Denham,2020He,2020VolkMalhotra}.\n\n\\section*{Acknowledgements}\n\nThis is a pre-copyedited, author-produced PDF of an article accepted for publication in MNRAS  following peer review. The version of record Marshall et al., 2020, MNRAS, 494, 2, 2280--2288 is available online \\href{https:\/\/academic.oup.com\/mnras\/article\/494\/2\/2280\/5819459}{here}.\n\nWe thank the anonymous referee for their comments which helped to improve the article.\n\nThis research has made use of NASA's Astrophysics Data System and the SIMBAD database, operated at CDS, Strasbourg, France.\n\nJPM acknowledges research support by the Ministry of Science and Technology of Taiwan under grants MOST104-2628-M-001-004-MY3 and MOST107-2119-M-001-031-MY3, and Academia Sinica under grant AS-IA-106-M03.\n\n\\textit{Software}: This research has made use of the following Python packages: \\textsc{matplotlib} \\citep{2007Hunter}; \\textsc{numpy} \\citep{2006Oliphant}; \\textsc{pygtc} \\citep{2016Bocquet}; \\textsc{emcee} \\citep{2013ForemanMackey}; \\textsc{corner} \\citep{2016ForemanMackey}; \\textsc{mercury} \\citep{1999Chambers}.\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{Motivation}\n\\IEEEPARstart{T}{he} emerging of many dazzling innovative technologies such as ultra-massive MIMO, large-scale intelligent reflective surfaces (IRS) and wireless artificial intelligence (AI) \\cite{Saad,Yang}, etc., boosts the rapid development of broadband wireless communications. On the other hand, in the foreseeable future, many revolutionary and challenging application scenarios such as smart city \\cite{Zanella}, autonomous driving \\cite{Toutouh} and unmanned aerial vehicle (UAV) positioning \\cite{Bor}, etc, may call for not only broadband connections but also accurate environment information which include but are not limited to, the locations, shapes, status and electromagnetic characteristic of the stationary or moving objects and\/or the background scatterers, within that environment.\n\nSuch kind of environment sensing has long been accomplished by traditional Radar technology which has been regarded as a related but separate field from communication. However, the channel state information (CSI) obtained during the communication process usually contains certain knowledge of the environment \\cite{Sen, Rao}, and likewise, the environment sensing result also helps improve the accuracy of channel estimation and enhance the performance of communication \\cite{Jiaoicc,Jiaotwc}. As the wireless network operates in higher frequency with wider bandwidth and deploys denser base stations with a larger amount of antennas, its longly-neglected sensing capability inherited from the intrinsic nature of electromagnetic wave propagation becomes even more explorable. The signal processing principles employed in these two fields also tend to converge. As visioned in \\cite{Wild}, this may give rise to a promising new technology of JCAS or sensing-communication integration.\n\nOne major challenge in JCAS lies in the potentially large amount of unknown variables brought by the environment, even many more than those contained in the statistical channel models, which may make the problem rank-deficient and eventually insolvable. As a result, certain sparsity should be exploited. Fortunately, the targeted environment itself usually does possess certain sparsity, which in turn makes the communication channel sparse. For example, in a cellular communication network, buildings are sparsely located within the wireless network coverage, and in the indoor scenario, furniture and other items are sparsely distributed in the entire room. \n\nAs a matter of fact, either in conventional communication or in conventional sensing, sparsity has been fully exploited to achieve better and lower-complexity solutions under the compressed sensing (CS) framework. As in wireless communications, the CS-based channel estimation approaches exploiting the channel sparsity usually exhibit superiority in signaling overhead and computational complexity \\cite{Rao} compared to the conventional channel estimation approaches \\textit{et al.}\\cite{Sen}. Likewise, in the broad area of Radar sensing or computational imaging, the utilization of the intrinsic sparsity of objects or scatterers within an environment is key to their effective detection. Such problems are often modeled as sparse signal recovery problems based on pixel division according to the CS theory \\cite{Donoho, Candes}, and solved by the widely used methods like Sparse Bayesian Learning (SBL) \\cite{Zhang}, orthogonal matching pursuit (OMP) \\cite{Cai}, and Generalized Approximate Message Passing (GAMP) \\cite{Rangan}, and so on. This has been showcased by recent works in \\cite{taoy, yaojj}, where innovative microwave computational imaging methods with the aid of intelligent reflecting surfaces (IRS) are proposed based on a fast block sparse Bayesian learning (BSBL) algorithm. However, how to explore and exploit the sparsity in the JCAS scenario still lacks adequate study.\n\n\\subsection{Related Works}\nSo far, there have been different sorts of attempts to implement joint environment sensing and communication. \n\nTo list a few, in the Radar-Communication Coexistence (RCC) sort of approaches \\cite{wang2008, Saruthirathanaworakun}, effective interference cancellation and management mechanism are designed to achieve flexible coexistence between the radar and the communication systems. In contrast to the RCC system, the second sort of approach, i.e., Dual-Functional Radar-Communication (DFRC), aim to achieve an integration of radar and communication through sharing a common hardware platform, with improved sensing and communication performance through collaborative operation \\cite{Paul, Blunt}. In such systems, separate sensing and communication operations with shared radio resources have also been extensively studied. For instance, a multi-beam scheme is proposed in \\cite{ZhangAndrew}, which uses an analog array to generate multiple beams for simultaneous communication and radar scanning. \n\nThe third sort of works mainly concentrates on the purpose of environment sensing by pure usage of the conventional communication signals or by proper joint design of the radio signals, which can be found in \\cite{AndrewEnabling} and the references therein. As in one of such works on environment sensing with the aid of real deployed communication systems, \\cite{Tan}  identified the behavior of a human body by extracting the Doppler frequency shift from the CSI conveyed by the WIFI communication signals in the indoor scenario. In \\cite{Daniels}, the authors used the orthogonal frequency division multiplexing communication waveform as a radar signal to achieve joint communication and environment sensing between vehicles. As an illustration of joint sensing and communication signaling, the authors in \\cite{ChenPerformance} designed a cooperative sensing unmanned aerial vehicle network (CSUN) with joint sensing and communication beams based on a common transceiver device. Considering the non-ideal factors of the channel, \\cite{Shahi} analyzed the communication channel capacity under the joint effect of Gaussian random noise and non-Gaussian radar sensing interference. In \\cite{Ahmadipour}, the authors theoretically analyzed the performance of the JCAS system under the condition of the memoryless broadcast channel. The related systematic design rules and methodologies for signaling and processing in such a JCAS system have raised increasing research effort recently, resulting in some interesting JCAS implementations based on machine learning \\cite{Aoudia}, joint data sensing and fusion \\cite{Schmitt}, and time-frequency-space full dimension utilization \\cite{Gaudio}, etc. \n\n\\subsection{Main Ideas and Contributions}\n\nIn this paper, we exploit the sparsity of both the structured multi-user signals and the unstructured environment to design a low-complexity joint multi-user communication and environment sensing scheme based on microwave computational imaging.\nDifferent from the above-mentioned application scenarios, we aim at designing a system with integrated sensing and communication capability based on existing wireless communication systems, which is capable of accomplishing the environment sensing (imaging) using the multi-user transmission signals and in turn, assisting the multi-user information detection with the channel information derived from the sensing results. To the best of our knowledge, there still lacks sufficient study on the design of such a JCAS system in the literature.\n\nIn order to achieve these goals, we employ the Sparse Code Multiple Access (SCMA) protocol \\cite{Nikopour,Taherzadeh} for multi-user uplink access, and employ an IRS \\cite{Garcia} to assist signal propagation and collection. SCMA is an elegant code-domain non-orthogonal multiple access (C-NOMA) method, which has extracted extensive research effort due to its superior performance and low detection complexity. In the SCMA scheme, the codebook for users to send data is sparse, and each user occupies a few but not all subcarriers. The sparsity of the user codebook effectively enhances the decoding performance of the received data. IRS is a promising technology to manipulate the electromagnetic environment with low-cost passive reflective elements by adjusting the phase of incident signals, which has been extensively used in wireless communications \\cite{Huang, Garcia, Wu, Chenw}.  Based on the idea of computational imaging, IRS can cause known and diverse changes to electromagnetic signals, and such changes are beneficial to environment sensing. Therefore, IRS also exhibited great potential in environment sensing as recently described in \\cite{yaojj, taoy}. But in these cases, the base station is only used as an environment sensing device instead of a communication signal transmitter. It is noteworthy that, in this work, the environment sensing is just accomplished by making use of the IRS' reflection characteristics rather than by actively manipulating it. Our method not only enables environment sensing to obtain the help of IRS but also retains the ability of IRS to assist communication.\n\nOur design is depicted as follows. First, in the multiple access part, the user uses the SCMA protocol to communicate with the wireless AP. The signals are reflected by the IRS and then arrive at the AP. With the limited channel information obtained by an initial pilot sequence, we use the proposed SCMA-IRS-MPA algorithm (see Section. \\ref{mainalgm} for details) to conduct multi-user detection based on the sparse codebook of the transmitted signals. Then a sliding-window-based environment sensing algorithm is proposed to accomplish the environment sensing (imaging) with the received signal and recovered users' data, again based on the CS principle. Note that the proposed multiple user detection algorithm requires the environment (channel) knowledge, and the proposed environment sensing algorithm also needs to know the decoded data, so the two processes, i.e., multiple access and environment sensing, rely on each other. Therefore, finally, we propose an iterative and incremental algorithm to jointly recover the users' data and accomplish environment sensing at the same time with significantly reduced pilot overhead.\n\nThe main contributions of this paper are summarized as follows:\n\n\\begin{itemize}\n    \\item We design a joint communication and environment sensing scheme, which exploits the sparsity of both the structured multi-user signals and the unstructured environment to achieve the integration of multiple access and environment sensing. \n    \n    \\item We develop a low-complexity iterative algorithm based on CS and generalized message passing theory to conduct the multi-user information detection and environment sensing (imaging). It is sliding-window based and runs alternately between the states of sensing with the decoded multi-user data and data decoding with the sensing results. This way, the overall system performance can be incrementally improved.   \n    \n    \\item We analyze the decoding error and sensing accuracy performances as well as the computational complexity of the proposed algorithm, and investigate the impact of access user number on system performances, base on which, we approximate the optimal operating point. Extensive simulation results verify the convergence and effectiveness of the proposed algorithm.\n        \n\\end{itemize}\n\nThe rest of this paper is organized as follows. Section \\uppercase\\expandafter{\\romannumeral2} presents the environment setting and system model in the uplink communication scenario. Section \\uppercase\\expandafter{\\romannumeral3} proposes the multiple access method based on the SCMA scheme. Section \\uppercase\\expandafter{\\romannumeral4} proposes the environment sensing method base on the CS theory. Section \\uppercase\\expandafter{\\romannumeral5} proposes the iterative and incremental algorithm based on low-density pilots jointly recovers the users' data and achieves environment sensing. In section \\uppercase\\expandafter{\\romannumeral6}, we discuss the trade-off relationship between the number of access users and system performance. Finally, section \\uppercase\\expandafter{\\romannumeral7} presents the numerical results, and section \\uppercase\\expandafter{\\romannumeral8} concludes the paper.\n\n\\textit{Notation}: Fonts $a$ and $\\mathbf{A}$ represent scalars and matrices, respectively.\n$\\mathbf{A}^{\\rm{T}}$ and $\\|\\mathbf{A}\\|_F$ denote transpose and Frobenius norm of $ \\mathbf{A} $, respectively.\n$[\\mathbf{A}](i,j)$ represents $\\mathbf{A}$'s $(i,j)$-th element.\n$|\\cdot|$ and $[\\cdot]$ denote the modulus and the catenation of the matrix, respectively.\n$\\odot $ represents the Hadamard product between two matrices.\nFinally, notation ${\\rm diag}(\\mathbf{a})$ represents a diagonal matrix with the entries of $\\mathbf{a}$ on its main diagonal, and $\\delta(\\cdot)$ is the Dirac delta function.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=3in]{fig1.eps}\n  \\caption{The uplink multi-user communication scenario.}\n  \\label{figsetting}\n  \\end{figure}\n\n\\section{Environment Setting and System Model}\nAs shown in Fig. \\ref{figsetting}, a millimeter-wave multi-antenna AP and multiple single-antenna user equipments (UEs) are deployed in an indoor scenario. An IRS is deployed near the AP to assist the users' communication, and there are some target objects (serve as scatterers) in the scenario. We consider that in the uplink regime, i.e., multiple users simultaneously send data to the AP via a shared channel. Our goal is to accomplish the environment sensing while reliably obtaining the communication data, that is, to sense the distribution position and scattering rate of the scatterers in the environment and obtain the communication data of all the users at the same time.\n\n\\subsection{Environment Setting}\nLet the number of users in the environment be $N_{\\rm{u}}$ and the number of AP receiving antennas be $N_{\\rm{R}}$. In the uplink communication scenario, the channel from the user to the AP has mainly composed of three parts: The first part is the line-of-sight (LOS) path from the user directly to the AP. The second part is the path from the user to the IRS and then reflected to the AP. The third part is the multipath propagation path which the user signal is scattered by the scatterers and reflected to the AP by the IRS. They are denoted as ${{\\bf{H}}^{{\\rm{LOS}}}} \\in \\mathbb{C}^{{N_{\\rm{u}}} \\times {N_{\\rm{R}}}}$,  ${{\\bf{H}}^{{\\rm{IRS}}}} \\in \\mathbb{C}^{{N_{\\rm{u}}} \\times {N_{\\rm{R}}}}$, and ${{\\bf{H}}^{{\\rm{s}}}} \\in \\mathbb{C}^{{N_{\\rm{u}}} \\times {N_{\\rm{R}}}}$ respectively. Let the number of reflective elements of the IRS be $N_{\\rm{I}}$, and each element can set amplitude reflection coefficient and phase shift independently, thereby controlling the relationship between the reflected signal and the incident signal. The reflection characteristic matrix of IRS is expressed as\n\\begin{equation}\n{\\bf{\\Theta }} = {\\rm{diag}}\\left( {{\\theta _1}, \\cdots, {\\theta _{{N_{\\rm{I}}}}}} \\right) \\in \\mathbb{C}^{{N_{\\rm{I}}} \\times {N_{\\rm{I}}}}, \\label{eq1}\n\\end{equation}\nwhere ${\\theta _{{n_{\\rm{I}}}}} = {\\rho _{{n_{\\rm{I}}}}}{e^{j{\\varphi _{{n_{\\rm{I}}}}}}}$ represents the reflection characteristic of the $n_{\\rm{I}}$ element of the IRS, ${\\rho _{{n_{\\rm{I}}}}} \\in \\left[ {0,1} \\right]$ and ${\\varphi _{{n_{\\rm{I}}}}} \\in \\left[ {{\\rm{0}},2\\pi } \\right]$ represents the amplitude reflection coefficient and phase shift of the $n_{\\rm{I}}$ element respectively, and the diag function represents the construction of a diagonal matrix.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=2in]{fig14.eps}\n  \\caption{The discretized targeted environment object.}\n  \\label{pixel}\n  \\end{figure}\n\nWe discretize the room environment information and regard the environment information of the entire room as a point cloud. Each point in the point cloud represents the environment information of small cubes with sizes $l_{\\rm{s}}$, $w_{\\rm{s}}$, and $h_{\\rm{s}}$ around this point. These small cubes are called pixels. Assuming that the length, width, and height of the room are $L_{\\rm{s}}$, $W_{\\rm{s}}$, and $H_{\\rm{s}}$ respectively, the number of point clouds in the space is ${N_{\\rm{s}}} = {{{L_{\\rm{s}}}} \\mathord{\\left\/\n{\\vphantom {{{L_{\\rm{s}}}} {{l_{\\rm{s}}}}}} \\right.\n\\kern-\\nulldelimiterspace} {{l_{\\rm{s}}}}} \\times {{{W_{\\rm{s}}}} \\mathord{\\left\/\n{\\vphantom {{{W_{\\rm{s}}}} {{w_{\\rm{s}}}}}} \\right.\n\\kern-\\nulldelimiterspace} {{w_{\\rm{s}}}}} \\times {{{H_{\\rm{s}}}} \\mathord{\\left\/\n{\\vphantom {{{H_{\\rm{s}}}} {{h_{\\rm{s}}}}}} \\right.\n\\kern-\\nulldelimiterspace} {{h_{\\rm{s}}}}}$. The inside of each pixel may be empty, or there may be scatterers. We use a scattering coefficient ${x_{{n_{\\rm{s}}}}}$ to represent the scattering coefficient of the pixel where the $n_{\\rm{s}}$-th point cloud point is located. If the inside of the small cube is empty, then ${x_{{n_{\\rm{s}}}}} = 0$. As shown in the Fig. \\ref{pixel}, the target scatterer in Fig. \\ref{figsetting} is discretized. Therefore, the environmental information of the entire room can be expressed as\n\\begin{equation}\n{\\bf{x}} = {\\left[ {{x_1},{x_2}, \\cdots ,{x_{{N_{\\rm{s}}}}}} \\right]^{\\rm{T}}}. \\label{eq2}\n\\end{equation}\n\n\\subsection{System Model}\nMultiple users in the space share the same time-frequency resources. The frequency resources of communication are divided into $R$ orthogonal resource elements (OREs), and $N_u$ users send data on all OREs. Therefore, on the $r$-th ORE, the channel from the user to the AP can be expressed as\n\\begin{equation}\n\\begin{array}{l}\n  {{\\bf{H}}_r} = {\\bf{H}}_r^{{\\rm{LOS}}} + {\\bf{H}}_r^{{\\rm{IRS}}} + {\\bf{H}}_r^{\\rm{s}}\\\\\n  \\quad  = {\\bf{H}}_r^{{\\rm{LOS}}} + {\\bf{H}}_r^{{\\rm{IRS1}}}{\\bf{\\Theta H}}_r^{\\rm{s1}}\\\\\n  \\quad  + \\left[ {\\begin{array}{*{20}{c}}\n  {{\\bf{H}}_r^{\\rm{s}}\\left(1\\right)} & \\cdots & {{\\bf{H}}_r^{\\rm{s}}\\left(n_{\\rm{u}}\\right)}& \\cdots &{{\\bf{H}}_r^{\\rm{s}}\\left(N_{\\rm{u}}\\right)}\n  \\end{array}} \\right]\n  \\end{array}, \\label{eq3}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\bf{H}}_r^{\\rm{s}}\\left(n_{\\rm{u}}\\right) = {{\\bf{x}}^{\\rm{T}}}{\\rm{diag}}\\left( {{\\bf{H}}_r^{\\rm{s3}}}\\left(n_{\\rm{u}}\\right) \\right){\\bf{H}}_r^{\\rm{s2}}{\\bf{\\Theta H}}_r^{\\rm{s1}} \\in \\mathbb{C}^{{\\rm{1}} \\times {N_{\\rm{R}}}} \\label{eq4}\n\\end{equation}\nrepresents the channel coefficient from the $n_u$-th user to the AP after being scattered by the scatterers, and\n${\\bf{H}}_r^{{\\rm{LOS}}} = {\\bm{\\alpha }}_r^{{\\rm{LOS}}}\\odot   {e^{j{\\bm{\\varphi }}_r^{{\\rm{LOS}}}}} \\in \\mathbb{C}^{{{N_{\\rm{u}}}} \\times {N_{\\rm{R}}}}$ represents the LOS channel coefficient from the user directly to the AP, with ${\\bm{\\alpha }}_r^{{\\rm{LOS}}}$ denoting the amplitude of the channel and ${e^{j{\\bm{\\varphi }}_r^{{\\rm{LOS}}}}}$ its phase shift. Similarly, ${\\bf{H}}_r^{{\\rm{IRS1}}} = {\\bm{\\alpha }}_r^{{\\rm{IRS1}}}\\odot {e^{j{\\bm{\\varphi }}_r^{{\\rm{IRS1}}}}} \\in \\mathbb{C}^{{{N_{\\rm{u}}}} \\times {N_{\\rm{I}}}}$, ${{\\bf{H}}_r^{{\\rm{s1}}}} = {\\bm{\\alpha }}_r^{\\rm{s1}}\\odot {e^{j{\\bm{\\varphi }}_r^{\\rm{s1}}}} \\in \\mathbb{C}^{{{N_{\\rm{I}}}} \\times {N_{\\rm{R}}}}$, ${\\bf{H}}_r^{\\rm{s3}}\\left(n_{\\rm{u}}\\right) = {\\bm{\\alpha }}_r^{\\rm{s3}}\\left(n_{\\rm{u}}\\right)\\odot {e^{j{\\bm{\\varphi }}_r^{{\\rm{s3}}}\\left(n_{\\rm{u}}\\right)}} \\in \\mathbb{C}^{{\\rm{1}} \\times {N_{\\rm{s}}}}$, ${\\bf{H}}_r^{{\\rm{s2}}} = {\\bm{\\alpha }}_r^{{\\rm{s2}}}\\odot {e^{j{\\bm{\\varphi }}_r^{{\\rm{s2}}}}} \\in \\mathbb{C}^{{{N_{\\rm{s}}}} \\times {N_{\\rm{I}}}}$ represent the LOS channel coefficient from the user to the IRS, from the IRS to the AP, from the user to the spatial point cloud location, and from the spatial point cloud location to the IRS, respectively.\n\nLet the $n_{\\rm{u}}$-th user's transmitted symbols on $R$ OREs be ${{\\bf{s}}_{{n_{\\rm{u}}}}} \\in \\mathbb{C}^{{{R}} \\times {1}}$, then at the $n_{\\rm{R}}$-th AP receiving antenna, the received data on all OREs can be expressed as\n\\begin{equation}\n{{\\bf{y}}_{{n_{\\rm{R}}}}} = \\sum\\limits_{{n_{\\rm{u}}} = 1}^{{N_{\\rm{u}}}} {{\\rm{diag}}\\left\\{ {{{\\bf{H}}\\left({n_{\\rm{u}}},{n_{\\rm{R}}}\\right)}} \\right\\}{{\\bf{s}}_{{n_{\\rm{u}}}}}}  + {\\bf{w}}, \\label{eq5}\n\\end{equation}\nwhere ${{\\bf{H}}\\left({n_{\\rm{u}}},{n_{\\rm{R}}}\\right)} = \\left[ {\\begin{array}{*{20}{c}}\n  {{{\\bf{H}}_1}\\left( {{n_u},{n_R}} \\right)}& \\cdots &{{{\\bf{H}}_R}\\left( {{n_u},{n_R}} \\right)}\n  \\end{array}} \\right]$ , ${\\bf{H}}_r\\left(n_{\\rm{u}},n_{\\rm{R}}\\right) $ represents the $n_{\\rm{R}}$ column and $n_{\\rm{u}}$ row of ${{\\bf{H}}_r}$ calculated in (\\ref{eq3}), and $\\bf{w}$ the Gaussian white noise.\n\n\\section{The Multiple Access Scheme}\nIn order to recover multi-user communication data accurately, the wireless access algorithm can be used to achieve the separation and detection of multi-user transmission symbols under the premise of environmental prior information.\n\n\\subsection{Sparse Code Multiple Access}\nSCMA is an efficient code-domain non-orthogonal multiple access technology. It is based on low-density spectrum spreading. That means, a single user does not completely occupy all OREs, but only a few of them, which greatly reduces the difficulty of signal decoding. In the uplink multi-user SCMA communication scenario considered in this article, the $N_{\\rm{u}}$ users use the SCMA protocol to send their data to the AP and $N_{\\rm{u}}$ users share $R$ OREs. Each user has a total of $M$ input possibilities, and each user accesses the channel by using a unique sparse codebook ${{\\bf{C}}_{{n_{\\rm{u}}}}} \\in \\mathbb{C}^{{{R}} \\times {M}}$. Therefore, each user's codebook contains $M$ codewords. Let ${{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m\\right)} \\in \\mathbb{C}^{{{R}} \\times {1}}$ represent the $m$-th codeword of user $n_{\\rm{u}}$. Then the (\\ref{eq5}) can be expressed as\n\\begin{equation}\n{{\\bf{y}}_{{n_{\\rm{R}}}}} = \\sum\\limits_{{n_{\\rm{u}}} = 1}^{{N_{\\rm{u}}}} {{\\rm{diag}}\\left\\{ {{{\\bf{H}}\\left({{n_{\\rm{u}}},{n_{\\rm{R}}}}\\right)}} \\right\\}{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m\\right)}}  + {\\bf{w}}, \\label{eq6}\n\\end{equation}\nwhere ${{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m\\right)}$ represents the sending symbols, it contains $d_{\\rm{v}}$ non-zero elements, that is, each user will only transmit on the OREs represented by $d_{\\rm{v}}$ non-zero elements. $N_{\\rm{u}}$ users perform overload transmission on all OREs, and the number of users $d_{\\rm{f}}$ transmitted on each ORE is constant. Since the codewords $\\bf{C}$ is sparse, not all users' codewords will collide on a single ORE. Fig. \\ref{figSCMA} shows an example of an uplink SCMA system, in which 6 users transmit on 4 OREs, thus $N_{\\rm{u}}$ = 6, $R$ = 4. Each user has its own codebook, and the codebook determines the OREs occupied by the user. In Fig. \\ref{figSCMA}, user 1 transmits on ORE 1 and 2, and user 2 transmits on ORE 3 and 4. The user's bitstream is mapped to the codewords by SCMA encoder after channel-coded, then transmitted to the receiver through the channel, and finally separated and decoded by the SCMA detector.\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=6in]{fig2.eps}\n  \\caption{The uplink SCMA system ($N_{\\rm{u}} = 6$, $R = 4$).}\n  \\label{figSCMA}\n  \\end{figure*}\n\nCodebook design is an important physical layer technology in the SCMA system. The sparsity of the codebook makes it possible for the SCMA receiver to use message passing algorithm (MPA) to decode. As mentioned above, the process of SCMA encoding is the process of mapping the binary bit stream to the complex domain. The codebook of each user is an ${R}\\times{M}$-dimensional matrix. Therefore, the SCMA encoder can be defined as: $f:\\mathbb{B}^{{{\\log }_2}M} \\to {\\cal X}$, where. ${\\cal X} \\subset \\mathbb{C} ^R,\\left| {\\cal X} \\right| = M$. Let $\\bf{b}$ represent the user's input bits, the corresponding codeword output can be expressed as ${\\cal X} = f\\left( {\\bf{b}} \\right)$, the codeword ${\\cal X}$ is an $R$-dimensional sparse complex vector, and the vector contains $N_{\\rm{c}} < R$ non-zero elements. Since SCMA encoding process combines bit-to-constellation mapping and spreading spectrum, the bit-to-constellation mapping can be expressed as: $g:\\mathbb{B} ^{{{\\log }_2}M} \\to {\\cal C},\\;\\;{\\cal C} \\subset \\mathbb{C} ^{N_{\\rm{c}}}$, where ${\\cal C}$ represents the constellation point of the $N_{\\rm{c}}$-dimensional complex constellation, so the SCMA encoder can also be expressed as : $f = {\\bf{V}}g$, where ${\\cal C} = f\\left( {\\bf{b}} \\right)$ and ${\\bf{V}} \\in \\mathbb{B} ^{R \\times N}$ is a binary mapping matrix, and the mapping matrix can map $N_{\\rm{c}}$-dimensional constellation points to $R$-dimensional SCMA codewords. Meanwhile, the mapping matrix of each user is different, and contains $R-N_{\\rm{c}} $ all-zero rows.\n\nDefine the codebook structure of SCMA as ${\\cal S}\\left( {{\\cal V},{\\cal G};{N_{\\rm{u}}},M,N_{\\rm{c}},R} \\right)$, where ${\\cal V}: = \\left[ {{{\\bf{V}}_{{n_{\\rm{u}}}}}} \\right]_{{n_{\\rm{u}}} = 1}^{{N_{\\rm{u}}}}$, ${\\cal G}: = \\left[ {{g_{{n_{\\rm{u}}}}}} \\right]_{{n_{\\rm{u}}} = 1}^{{N_{\\rm{u}}}}$. Therefore, the SCMA codebook design problem can be expressed as\n\\begin{equation}\n{{\\cal V}^*},{{\\cal G}^*} = \\arg \\mathop {\\max }\\limits_{{\\cal V},{\\cal G}} {\\cal M}\\left( {{\\cal S}\\left( {{\\cal V},{\\cal G};{N_{\\rm{u}}},M,N_{\\rm{c}},R} \\right)} \\right), \\label{eq7}\n\\end{equation}\nwhere ${\\cal M}$ is a codebook design standard. Since there is no unified design standard at present, there are many methods for SCMA codebook design problems, such as rearranging the real and imaginary parts of the constellation points and designing codebooks based on theories such as constellation interleaving and rotation. These methods can achieve a suboptimal solution to the SCMA codebook design problem.\n\n\\subsection{SCMA-IRS-MPA Decoder}\\label{mainalgm}\nIn the above-mentioned SCMA-IRS uplink transmission scheme, there are a total of ${M^{{N_{\\rm{u}}}}}$ combinations of user codewords. The Maximum Likelihood (ML) decoder can provide the theoretically optimal symbol error rate (SER) performance by performing a traversal search on all codewords combinations. The estimated transmit codewords of all users by the ML decoder can be expressed as\n\\begin{equation}\n\\begin{array}{l}\n  {{{\\bf{\\hat C}}}_{{\\rm{ML}}}} = \\arg \\mathop {\\min }\\limits_{j \\in {M^{{n_{\\rm{u}}}}}} \\\\\n  \\quad \\quad {\\left\\| {{{\\bf{y}}_{{n_{\\rm{R}}}}} - \\sum\\limits_{{n_{\\rm{u}}} = 1}^{{N_{\\rm{u}}}} {\\left( {{\\rm{diag}}\\left( {{\\bf{H}}\\left( {{n_{\\rm{u}}},{n_{\\rm{R}}}} \\right)} \\right){{\\bf{C}}_{{n_{\\rm{u}}}}}\\left( {{\\bf{m}}\\left( j \\right)} \\right)} \\right)} } \\right\\|^2}\n  \\end{array}, \\label{eq8}\n\\end{equation}\nwhere ${{\\bf{\\hat C}}_{\\rm{ML}}} = \\left[ {{{{\\bf{\\hat c}}}_{1}}, \\cdots ,{{{\\bf{\\hat c}}}_{{N_{\\rm{u}}}}}} \\right] \\in \\mathbb{C} ^{R \\times {N_{\\rm{u}}}}$, ${\\bf{m}}(j)$ represents the value of the $j$-th combination among ${M^{{N_{\\rm{u}}}}}$ user codewords combinations. Although the ML decoder can provide the theoretical optimal value, it uses an exhaustive method to search for the optimal solution, which is impractical in the actual implementation process. MPA decoder is an iterative decoder, which can nearly achieve the performance of an ML decoder while requiring an achievable computational complexity. And the MPA decoder obtains the corresponding user codewords by calculating the maximum joint message probability.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=7cm]{fig3.eps}\n  \\caption{The MPA decoder factor graph ($N_{\\rm{u}} = 6$, $R = 4$).}\n  \\label{figFG}\n  \\end{figure}\n\nMPA is a belief propagation algorithm that uses a factor graph model to solve probabilistic reasoning problems. The proposed SCMA-IRS-MPA uses the factor graph method shown in the Fig. \\ref{figFG}, where the function nodes (FNs) represent OREs, the variable nodes (VNs) represent the users, and the connection between the FN and VN represents the user transmitting data on the corresponding ORE. The MPA decoder achieves decoding by iteratively updating the message probability between FNs and VNs, and let the MPA decoder stop after ${K_{\\rm{it}}}$ iterations. In order to estimate the transmission codewords in the SCMA-IRS-MPA scheme, we modified the traditional SCMA-MPA. In our method, multiple antennas at the AP perform independent and parallel decoding. For the $n_{\\rm{R}}$-th receiving antenna, we use $p_{{v_u} \\to {f_r}}^{\\left( {{k_{\\rm{it}}}} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)$ to denote the probability of transmitting a message from the $n_{\\rm{u}}$-th VN to the $r$-th FN, and use $p_{{f_r} \\to {v_u}}^{\\left( {{k_{\\rm{it}}}} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)$ to denote the probability of transmitting a message from the $r$-th FN to the $n_{\\rm{u}}$-th VN. The above all represent the probability in the ${K_{\\rm{it}}}$ round iteration, ${k_{\\rm{it}}} = 1,2, \\cdots ,{K_{\\rm{it}}}$. Assuming that at the beginning, in the first iteration, all messages sent from VN to FN have the same probability,\n\\begin{equation}\np_{{v_u} \\to {f_r}}^{\\left( 0 \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right) = \\frac{1}{M},\\left( {\\forall {n_{\\rm{u}}},\\forall r,\\forall m} \\right). \\label{eq9}\n\\end{equation}\n\nTherefore, $p_{{f_r} \\to {v_u}}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)$ can be expressed as\n\\begin{equation}\n\\begin{array}{l}\n  p_{{f_r} \\to {v_u}}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right){\\rm{ = }}\\\\\n  \\quad \\quad \\sum\\limits_{\\psi \\left( i \\right),i \\in {{\\bf{\\Lambda }}_r}\\backslash {n_{\\rm{u}}}} {\\left\\{ {p\\left( {{\\bf{y}}|\\psi \\left( i \\right),\\psi \\left( u \\right) = {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}}} \\right)} \\right.} \\\\\n  \\quad \\quad \\left. { \\times \\prod\\limits_{i \\in {{\\bf{\\Lambda }}_r}\\backslash {n_{\\rm{u}}}} {p_{{v_i} \\to {f_r}}^{\\left( {{k_{\\rm{it}}}} \\right)}} \\left( {\\psi \\left( i \\right)} \\right)} \\right\\},\\left( {\\forall m,\\forall r,{n_{\\rm{u}}} \\in {{\\bf{\\Lambda }}_r}} \\right)\n  \\end{array}, \\label{eq10}\n\\end{equation}\nwhere ${{\\bf{\\Lambda }}_r}$ represents a set of user indexes sharing the $r$-th ORE, ${{\\bf{\\Lambda }}_r}\\backslash {n_{\\rm{u}}}$ represents ${{\\bf{\\Lambda }}_r}$ except for the $n_{\\rm{u}}$-th user, and\n\\begin{equation}\n\\begin{array}{l}\n  p\\left( {{\\bf{y}}|{{\\bf{\\Psi }}_r}} \\right) = \\frac{1}{{\\sqrt {2\\pi } \\sigma }}\\exp \\left( { - \\left| {{{\\bf{y}}_r} - \\sum\\nolimits_{{n_{\\rm{u}}} \\in {{\\bf{\\Lambda }}_r}} {\\left( {{\\bf{H}}_r^{{\\rm{LOS}}}\\left( {{n_{\\rm{u}}},{n_{\\rm{R}}}} \\right)} \\right.} } \\right.} \\right.\\\\\n   \\quad \\left. {{{{{\\left. {\\left. { + {\\bf{H}}_r^{{\\rm{IRS}}}\\left( {{n_{\\rm{u}}},{n_{\\rm{R}}}} \\right) + {\\bf{H}}_r^{\\rm{s}}\\left( {{n_{\\rm{u}},n_{\\rm{R}}}} \\right)} \\right){{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right|}^2}} \/ {\\left( {2{\\sigma ^2}} \\right)}}} \\right)\n  \\end{array}, \\label{eq11}\n\\end{equation}\nwhere ${{\\bf{\\Psi }}_r}$ represents the possible codewords of all users sharing the $r$-th ORE, then $p_{{v_u} \\to {f_r}}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)$ is updated to,\n\\begin{equation}\n\\begin{array}{l}\n  p_{{v_u} \\to {f_r}}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right) = \\gamma _{{v_u},r}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\\\\n  \\quad \\quad \\quad  \\times \\prod\\limits_{j \\in {\\Omega _u}\\backslash r} {p_{{f_r} \\to {v_u}}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)} ,\\forall m,\\forall {n_{\\rm{u}}},r \\in {{\\bf{\\Omega }}_u}\n  \\end{array}, \\label{eq12}\n\\end{equation}\nwhere ${{\\bf{\\Omega }}_u}$ represents the ORE index corresponding to the $d_{\\rm{v}}$ non-zero element positions of the codeword of the $n_{\\rm{u}}$-th user, ${{\\bf{\\Omega }}_u}\\backslash r$ represents ${{\\bf{\\Omega }}_u}$ except for the $r$-th ORE, and $\\gamma _{{v_u},r}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}$ can be expressed as\n\\begin{equation}\n\\gamma _{{v_u},r}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}{\\rm{ = }}{\\left( {\\sum\\limits_{m = 1}^M {p_{{v_u} \\to {f_r}}^{\\left( {{k_{\\rm{it}}}} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)} } \\right)^{ - 1}}. \\label{eq13}\n\\end{equation}\n\nAfter $K_{\\rm{it}}$ iterations, the estimated transmission codewords of the $n_{\\rm{u}}$-th user can be expressed as\n\\begin{equation}\n{{{\\bf{\\hat C}}}_{{n_{\\rm{u}}}} ^{\\left( {{k_{it}}} \\right)}} = \\arg \\mathop {\\max }\\limits_{m = 1, \\cdots M} \\prod\\limits_{j \\in {\\Omega _u}} {p_{{f_j} \\to {v_u}}^{\\left( {{k_{\\rm{it}}}} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)} ,\\forall n_{\\rm{u}}. \\label{eq14}\n\\end{equation}\n\nThe set of all user transmission codewords obtained by using the SCMA-IRS-MPA decoder is\n\\begin{equation}\n{{\\bf{\\hat C}}_{{\\rm{MPA}}}} = \\left\\{ {{{ {{{{\\bf{\\hat c}}}_1}} }^{\\left( {{k_{\\rm{it}}}} \\right)}}, \\cdots ,{{ {{{{\\bf{\\hat c}}}_{{N_{\\rm{u}}}}}} }^{\\left( {{k_{\\rm{it}}}} \\right)}}} \\right\\}. \\label{eq15}\n\\end{equation}\n\nThe above is the MPA decoder of the SCMA-IRS scheme. We express the decoding computational complexity of the MPA decoder according to the number of addition operations and the number of multiplication operations. Therefore, the number of additions and multiplications required by the MPA detector are $R{d_{\\rm{f}}}\\left( {{M^{{d_{\\rm{f}}}}}\\left( {4{d_{\\rm{f}}} + {K_{\\rm{it}}} + 1} \\right) + N - {K_{\\rm{it}}}} \\right) + 1$ and $R{d_{\\rm{f}}}\\left( {{M^{{d_{\\rm{f}}}}}\\left( {4{d_{\\rm{f}}} + {K_{\\rm{it}}}{d_{\\rm{f}}} + 3} \\right) + N + M{K_{\\rm{it}}}\\left( {{d_{\\rm{v}}} - 1} \\right)} \\right) + {N_{\\rm{u}}}M\\left( {{d_{\\rm{v}}} - 1} \\right)$ respectively.\n\n\\section{Environment Sensing}\nContrary to the multiple access process, the algorithm proposed in this section can sense the environmental information with the data sent by the user has been decoded correctly. Since the distribution of scatterers in the environment is sparse, sensing environmental information is essential to solve the CS reconstruction problem. As shown in (\\ref{eq5}), on the $r$-th ORE, the solution of environmental information can be expressed as\n\\begin{equation}\n{\\bf{\\hat x}} = \\arg \\mathop {\\min }\\limits_{\\bf{x}} {\\left\\| {\\bf{x}} \\right\\|_1}\\quad \\quad {\\rm{s}}{\\rm{.t}}{\\rm{.}}\\quad {\\left\\| {{{\\bf{y}}_r} - {{\\bf{s}}_r}{{\\bf{H}}_r}} \\right\\|_2} \\le {\\varepsilon _{\\rm{x}}}, \\label{eq16}\n\\end{equation}\nwhere $\\varepsilon _{\\rm{x}}$ is the slack variable, ${{\\bf{y}}_r} \\in \\mathbb{C}^{{N_{\\rm{T}}} \\times {N_{\\rm{R}}}}$ is the symbol sequence received by the AP receiving antennas, $N_{\\rm{T}}$ is the time sequence length, and ${{\\bf{s}}_r} \\in \\mathbb{C} ^{{N_{\\rm{T}}} \\times {N_{\\rm{u}}}}$ is the transmitted symbol sequence of $N_{\\rm{u}}$ users. In the received signal model, when both the transmitted data ${\\bf{s}}_r$ and the received data ${\\bf{y}}_r$ are known, as shown in (\\ref{eq6}), the channel coefficients ${{\\bf{H}}\\left({{n_{\\rm{u}}},{n_R}}\\right)}$ can be obtained by simply solving the linear equations. After performing the same analysis on all the receiving antennas of the AP, the channel coefficient ${\\bf{H}}_r$ on the $r$-th ORE is solved.\n\nAs shown in (\\ref{eq3}), the ${\\bf{H}}_r^{{\\rm{LOS}}}$ and ${\\bf{H}}_r^{{\\rm{IRS}}}$ in the channel coefficient ${\\bf{H}}_r$ are composed of LOS channels. Meanwhile, the reflection characteristic matrix $\\bf{\\Theta} $ of the IRS used for assist communication is given, and only ${\\bf{H}}_r^{\\rm{s}}$ contains unknown environmental information. The $n_{\\rm{u}}$-th row of ${\\bf{H}}_r^{\\rm{s}}$ is expressed as\n\\begin{equation}\n{\\bf{H}}_r^{\\rm{s}}\\left(n_{\\rm{u}}\\right) = {{\\bf{x}}^{\\rm{T}}}{\\rm{diag}}\\left( {{\\bf{H}}_r^{{\\rm{s3}}}} \\left(n_{\\rm{u}}\\right) \\right){\\bf{H}}_r^{{\\rm{s2}}}{\\bf{\\Theta H}}_r^{{\\rm{s1}}}, \\label{eq17}\n\\end{equation}\n\\begin{equation}\n{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(n_{\\rm{u}}\\right)\\right)^{\\rm{T}}} = {{\\bf{A}}_r^{\\rm{s}}}\\left(n_{\\rm{u}}\\right){\\bf{x}}, \\label{eq18}\n\\end{equation}\nwhere ${{\\bf{A}}_r^{\\rm{s}}}\\left(n_{\\rm{u}}\\right) \\in \\mathbb{C}^{{N_{\\rm{R}}} \\times {N_{\\rm{s}}}}$ is the known channel coefficient, which is also called the measurement matrix in the CS problem. For $N_{\\rm{u}}$ users, the (\\ref{eq18}) is expressed as the matrix form of the CS problem,\n\\begin{equation}\n{\\left[ {\\begin{array}{*{20}{c}}\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(1\\right) \\right)}^{\\rm{T}}}}\\\\\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(2\\right) \\right)}^{\\rm{T}}}}\\\\\n   \\vdots \\\\\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(N_{\\rm{u}}\\right) \\right)}^{\\rm{T}}}}\n  \\end{array}} \\right]_{{N_{\\rm{u}}}{N_{\\rm{R}}} \\times 1}} = {\\left[ {\\begin{array}{*{20}{c}}\n  {{{\\bf{A}}_r^{\\rm{s}}}}\\left(1\\right)\\\\\n  {{{\\bf{A}}_r^{\\rm{s}}}}\\left(2\\right)\\\\\n   \\vdots \\\\\n  {{{\\bf{A}}_r^{\\rm{s}}}}\\left(N_{\\rm{u}}\\right)\n  \\end{array}} \\right]_{{N_{\\rm{u}}}{N_{\\rm{R}}} \\times {N_{\\rm{s}}}}}{\\left[ {\\bf{x}} \\right]_{{N_{\\rm{s}}} \\times 1}}\n \\end{equation} \n\\begin{equation}\n\\Rightarrow  {{\\bf{\\tilde{H}}}_r^{\\rm{s}}}  = {\\bf{\\tilde{A}}}_r^{\\rm{s}}{\\bf{x}}. \\label{eq19}\n\\end{equation}\n\n\\subsection{Generalized Approximate Message Passing}\nThe GAMP Algorithm\\cite{Rangan} solves the problem of CS sparse reconstruction by iterative decomposition. The above problem formula (\\ref{eq19}) is abbreviated as ${\\bf{y}} = {\\bf{\\Phi x}} + {\\bf{w}}$, where ${\\bf{\\Phi }} \\in \\mathbb{C} ^{{M_\\phi } \\times {N_\\phi }}$ is the CS measurement matrix and ${\\bf{w}} \\sim {\\cal C}{\\cal N}\\left( {0,{\\sigma ^{\\rm{w}}}} \\right)$ represents noise. In this article, we assume that the distribution of environmental scatterers information as a Bernoulli-Gaussian distribution in a limited interval which probability density function is expressed as\n\\begin{equation}\n\\begin{array}{l}\n  {p_{X{\\rm{|}}{\\bf{Q}}}}\\left( {x|{\\bf{q}}} \\right) = \\left( {1 - \\lambda  + \\alpha } \\right)\\delta \\left( x \\right)\\\\\n  \\quad \\quad \\quad  + \\lambda {\\cal N}\\left( {x|\\theta ,{\\sigma ^{\\rm{x}}}} \\right)\\left[ {u\\left( x \\right) - u\\left( {x - 1} \\right)} \\right]\n  \\end{array}, \\label{eq20}\n\\end{equation}\nwhere all parameters be expressed as ${\\bf{q}} \\buildrel \\Delta \\over = \\left[ {\\lambda ,\\alpha ,\\theta ,{\\sigma ^{\\rm{x}}}} \\right]$, $\\delta \\left(  \\cdot  \\right)$ is the Dirac function, $\\lambda $ is the sparsity coefficient, $\\alpha {\\rm{ = }}\\int_{x \\in \\left( { - \\infty ,0} \\right] \\cup \\left[ {1, + \\infty } \\right)} {\\lambda {\\cal N}\\left( {x|\\theta ,{\\sigma ^{\\rm{x}}}} \\right)} dx$. $\\theta  \\in \\left[ {{\\rm{0}},{\\rm{1}}} \\right]$ and ${\\sigma ^{\\rm{x}}}$ represent the mean and variance of the environmental scatterers information distribution respectively.\n\nThe GAMP algorithm has defined two parameterized functions ${g_{\\rm{in}}}\\left(  \\cdot  \\right)$ and ${g_{\\rm{out}}}\\left(  \\cdot  \\right)$ and the specific algorithm is shown in Algorithm 1. At this point, we will show how to specify the parameterized functions ${g_{\\rm{in}}}\\left(  \\cdot  \\right)$, ${g_{\\rm{out}}}\\left(  \\cdot  \\right)$, ${g'_{\\rm{in}}}\\left(  \\cdot  \\right)$ and ${g'_{\\rm{out}}}\\left(  \\cdot  \\right)$, based on the maximum posterior probability (MAP) estimation, the input function can be written as\n\\begin{equation}\n{g_{\\rm{in}}}\\left( {\\hat v,{\\sigma ^{\\rm{v}}},{\\bf{q}}} \\right) = \\arg \\mathop {\\max }\\limits_x {F_{\\rm{in}}}\\left( {x,\\hat v,{\\sigma ^{\\rm{v}}},{\\bf{q}}} \\right), \\label{eq22}\n\\end{equation}\n\\begin{equation}\n{F_{\\rm{in}}}\\left( {x,\\hat v,{\\sigma ^{\\rm{v}}},{\\bf{q}}} \\right) = \\log {p_{X|{\\bf{Q}}}}\\left( {x|{\\bf{q}}} \\right) - \\frac{1}{{2{\\sigma ^{\\rm{v}}}}}{\\left( {\\hat v - x} \\right)^2}, \\label{eq23}\n\\end{equation}\n\\begin{equation}\n{g'_{\\rm{in}}}\\left( {\\hat v,{\\sigma ^{\\rm{v}}},{\\bf{q}}} \\right) = \\frac{\\partial {g_{\\rm{in}}}\\left( {\\hat v,{\\sigma ^{\\rm{v}}},{\\bf{q}}} \\right)}{\\partial \\hat v} = \\frac{1}{{1 - {\\sigma ^{\\rm{v}}}\\frac{\\partial ^2}{\\partial  x^2} {\\rm log}\\left[{p_{X|{\\bf{Q}}}}\\left( {x|{\\bf{q}}} \\right)\\right]}}, \\label{eq24}\n\\end{equation}\nthe output function can be expressed as\n\\begin{equation}\n{g_{\\rm{out}}}\\left( {y,\\hat p,{\\sigma ^{\\rm{z}}}} \\right) = \\frac{{y - \\hat p}}{{{\\sigma ^{\\rm{w}}}} + {\\sigma ^{\\rm{z}}}}, \\label{eq25}\n\\end{equation}\n\\begin{equation}\n{g'_{\\rm{out}}}\\left( {y,\\hat p,{\\sigma ^{\\rm{z}}}} \\right) = \\frac{\\partial {g'_{\\rm{out}}}\\left( {y,\\hat p,{\\sigma ^{\\rm{z}}}} \\right)}{\\partial y} =  - \\frac{1}{{{\\sigma ^{\\rm{w}}}} + {\\sigma ^{\\rm{z}}}}. \\label{eq26}\n\\end{equation}\n\n\\begin{algorithm}[htb]\n  \\caption{The GAMP Algorithm\\cite{Rangan}}\n \n  \\begin{algorithmic}[1]\n  \\REQUIRE\n  Given measurement matrix ${\\bf{\\Phi }} \\in {\\mathbb{C}^{{M_\\phi } \\times {N_\\phi }}}$ and sequence of measurement value ${\\bf{y}} \\in \\mathbb{C} ^{{M_\\phi } \\times 1}$.\n  \\STATE\n  \\textbf{Initialization}: Set environment prior parameter $\\bf{q}$. Defined ${g_{\\rm{in}}}\\left(  \\cdot  \\right)$ and ${g_{\\rm{out}}}\\left(  \\cdot  \\right)$ from (\\ref{eq22}), (\\ref{eq24}). Set $t_i = 0$, ${\\bf{\\hat s}}\\left( { - 1} \\right) = 0$, ${\\hat x_{{n_\\phi }}}\\left( {{t_i}} \\right) > 0$, $\\sigma _{{n_\\phi }}^{\\rm{x}}\\left( {{t_i}} \\right) > 0$.\n\n  \\WHILE {$\\sum\\limits_{{m_\\phi }} {\\left( {{y_{{m_\\phi }}} - {{\\hat z}_{{m_\\phi }}}\\left( {{t_i}} \\right)} \\right)}  > \\varepsilon_{\\rm{t}} $, where $\\varepsilon_{\\rm{t}} $ is a given error tolerance value}\n  \\STATE\n  For each $m_\\phi $:\n\n  $\\sigma _{{m_\\phi }}^{\\rm{z}}\\left( {{t_i}} \\right) = \\sum\\limits_{{n_\\phi }} {\\Phi _{{m_\\phi },{n_\\phi }}^2} \\sigma _{{n_\\phi }}^{\\rm{x}}\\left( {{t_i}} \\right),$\n\n  ${\\hat p_{m_\\phi }}\\left( {t_i} \\right) = \\sum\\limits_{n_\\phi } {\\Phi _{{m_\\phi },{n_\\phi }}}{{\\hat x}_{n_\\phi }}\\left( {t_i} \\right) - \\sigma_{m_\\phi }^{\\rm{z}}\\left( t \\right) {\\hat s_{{m_\\phi }}}\\left( {{t_i} - 1} \\right),$\n\n  ${\\hat z_{{m_\\phi }}}\\left( {{t_i}} \\right){\\rm{ = }}\\sum\\limits_{{n_\\phi }} {{\\Phi _{{m_\\phi },{n_\\phi }}}} {\\hat x_{{n_\\phi }}}\\left( {{t_i}} \\right).$\n  \\STATE\n  For each $m_\\phi $:\n\n  ${\\hat s_{{m_\\phi }}}\\left( {{t_i}} \\right) = {g_{\\rm{out}}}\\left( {{t_i},{y_{{m_\\phi }}},{{\\hat p}_{{m_\\phi }}}\\left( {{t_i}} \\right),\\sigma _{{m_\\phi }}^{\\rm{z}}\\left( {{t_i}} \\right)} \\right),$\n\n  $\\sigma _{{m_\\phi }}^{\\rm{s}}\\left( {{t_i}} \\right) =  - {g'_{\\rm{out}}}\\left( {{t_i},{y_{{m_\\phi }}},{{\\hat p}_{{m_\\phi }}}\\left( {{t_i}} \\right),\\sigma _{{m_\\phi }}^{\\rm{z}}\\left( {{t_i}} \\right)} \\right).$\n  \\STATE\n  For each $n_\\phi $:\n\n  $\\sigma _{{n_\\phi }}^{\\rm{v}}\\left( {{t_i}} \\right) = {\\left[ {\\sum\\limits_{{n_\\phi }} {\\Phi _{{m_\\phi },{n_\\phi }}^2\\sigma _{{n_\\phi }}^{\\rm{s}}\\left( {{t_i}} \\right)} } \\right]^{ - 1}},$\n\n  ${\\hat v_{{n_\\phi }}}\\left( {{t_i}} \\right) = {\\hat x_{{n_\\phi }}}\\left( {{t_i}} \\right) + \\sigma _{{n_\\phi }}^{\\rm{v}}\\left( {{t_i}} \\right)\\sum\\limits_{{m_\\phi }} {{\\Phi _{{m_\\phi },{n_\\phi }}}{{\\hat s}_{{m_\\phi }}}\\left( {{t_i}} \\right)}.$\n  \\STATE\n  For each $n_\\phi $:\n\n  ${\\hat x_{{n_\\phi }}}\\left( {{t_i}{\\rm{ + 1}}} \\right) = {g_{\\rm{in}}}\\left( {{t_i},{{\\hat v}_{{n_\\phi }}}\\left( {{t_i}} \\right),\\sigma _{{n_\\phi }}^{\\rm{v}}\\left( {{t_i}} \\right),{\\bf{q}}} \\right),$\n\n  $\\sigma _{{n_\\phi }}^{\\rm{x}}\\left( {{t_i}{\\rm{ + 1}}} \\right) = \\sigma _{{n_\\phi }}^{\\rm{v}}\\left( {{t_i}} \\right){g'_{\\rm{in}}}\\left( {{t_i},{{\\hat v}_{{n_\\phi }}}\\left( {{t_i}} \\right),\\sigma _{{n_\\phi }}^r\\left( {{t_i}} \\right),{\\bf{q}}} \\right).$\n  \\STATE\n  ${t_i} = {t_i} + 1.$\n  \\ENDWHILE\n  \\ENSURE\n  Estimated sparse vector ${\\hat x_{{n_\\phi }}}\\left( {{t_i}} \\right)$ and $\\sigma _{{n_\\phi }}^{\\rm{x}}\\left( {{t_i}} \\right)$.\n  \\end{algorithmic}\n  \\end{algorithm}\n\n\\subsection{Proposed Environment Sensing Algorithm}\nIn the process of solving the CS sparse reconstruction problem, the product $N_{\\rm{u}}N_{\\rm{R}}$ of the number of users and the number of receiving antennas, and the number of spatial pixels $N_{\\rm{s}}$ are orders of magnitude different, that is ${N_{\\rm{u}}}{N_{\\rm{R}}} \\ll {N_{\\rm{s}}}$, so that the number of columns in the CS measurement matrix ${\\bf{\\tilde{A}}}_r^{\\rm{s}}$ is much larger than the number of rows, and the compression ratio is too high. Therefore, the environmental information $\\bf{x}$ cannot be recovered accurately. We improve the above algorithm to adapt to the continuous data stream sent from users in the proposed scenario. We use multiple data packets to recover the environmental information after multiple observations. The CS problem is redefined as\n\\begin{equation}\n{\\bf{H}}_r^{\\rm{s}}\\left(n_{\\rm{u}},k\\right) = {{\\bf{x}}^{\\rm{T}}}{\\rm{diag}}\\left( {{\\bf{H}}_r^{{\\rm{s3}}}}\\left(n_{\\rm{u}}\\right) \\right){\\bf{H}}_r^{{\\rm{s2}}}{{\\bf{\\Theta }}\\left(k\\right)}{\\bf{H}}_r^{{\\rm{s1}}}, \\label{eq27}\n\\end{equation}\n\\begin{equation}\n{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(n_{\\rm{u}},k\\right) \\right)^{\\rm{T}}} = {\\bf{A}}_r^{\\rm{s}}\\left(n_{\\rm{u}},k\\right){\\bf{x}}, \\label{eq28}\n\\end{equation}\n\\begin{equation}\n{\\left[ {\\begin{array}{*{20}{c}}\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(1,1\\right) \\right)}^{\\rm{T}}}}\\\\\n   \\vdots \\\\\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(n_{\\rm{u}},k\\right) \\right)}^{\\rm{T}}}}\\\\\n   \\vdots \\\\\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(N_{\\rm{u}},K\\right) \\right)}^{\\rm{T}}}}\n  \\end{array}} \\right]} = {\\left[ {\\begin{array}{*{20}{c}}\n  {{\\bf{A}}_r^{\\rm{s}}}\\left(1,1\\right)\\\\\n   \\vdots \\\\\n  {{\\bf{A}}_r^{\\rm{s}}}\\left(n_{\\rm{u}},k\\right)\\\\\n   \\vdots \\\\\n  {{\\bf{A}}_r^{\\rm{s}}}\\left(N_{\\rm{u}},K\\right)\n  \\end{array}} \\right]} \\left[{\\bf{x}}\\right] , \\label{eq29}\n\\end{equation}\n\\begin{equation}\n\\Rightarrow  \\left[{{\\bf{\\tilde{H}}}_r^{\\rm{s}}}\\left(K\\right)\\right]_{{N_{\\rm{u}}}{N_{\\rm{R}}}K \\times 1}  = \\left[{\\bf{\\tilde{A}}}_r^{\\rm{s}}\\left(K\\right)\\right]_{{N_{\\rm{u}}}{N_{\\rm{R}}}K \\times {N_{\\rm{s}}}}\\left[{\\bf{x}}\\right]_{{N_{\\rm{s}}}\\times 1}, \\label{eq30}\n\\end{equation}\nwhere ${\\bf{\\Theta }}\\left(k\\right)$ represents the IRS reflection characteristic matrix when the $k$-th data packet is received, and $K$ is the number of data packets. After multiple observations, the difference between the number of rows of the observation matrix $N_{\\rm{u}}N_{\\rm{R}}K$ and the number of columns $N_{\\rm{s}}$ is relatively small, and environmental information can be sensed more accurately.\n\nSince data packets are continuously transmitted during the communication process, and the amount of data is very large, it is impossible to store all the data packets $K$ sent at all times. We set a time sliding-window with a length of $n_{\\rm{f}}$, store the received data packet $k$ at the current moment to the $n_{\\rm{f}}$ data packets previously received, and use the data in the sliding-window for environment sensing.\n\nCompared with the communication data that exists all the time, the data in the sliding-window is very limited. Therefore, a large amount of communication data transmitted earlier will be wasted in the process of environment sensing. To solve this problem, we propose a ``momentum-mode'', which combines the sensing results calculated at the previous moments to calculate the current environment sensing results and makes the current sensing result contain part of the information outside the sliding-window. According to (\\ref{eq16}), (\\ref{eq18}), the ``momentum-mode'' can be expressed as\n\\begin{equation}\n{{{\\bf{\\hat x}}}_k}{\\rm{ = }}\\arg \\mathop {\\min }\\limits_{\\bf{x}} \\left( {{{\\left\\| {\\bf{x}} \\right\\|}_1}} \\right){\\rm{ + }}\\mu {{{\\bf{\\hat x}}}_{k{\\rm{ - 1}}}}, \\label{eq31}\n\\end{equation}\n\\begin{equation}\n{\\rm{s}}{\\rm{.t}}{\\rm{.}}\\quad {\\left\\|{ {{\\bf{\\tilde{H}}}_r^{\\rm{s}}}\\left(k\\right) - {\\bf{\\tilde{A}}}_r^{\\rm{s}}\\left(k\\right)}{\\bf{x}} \\right\\|_2} \\le {\\varepsilon _{\\rm{x}}}, \\label{eq32}\n\\end{equation}\nwhere $\\mu $ is the momentum coefficient. The larger the momentum coefficient $\\mu $, the more previous data information the current sensing result depends on. Therefore, the setting of the momentum coefficient $\\mu $ should be set according to the actual system and will be further analyzed in section \\uppercase\\expandafter{\\romannumeral7}.\n\n\\section{Joint Multi-User Detection and Environment Sensing Algorithm}\nAs mentioned in Section \\uppercase\\expandafter{\\romannumeral3} and Section \\uppercase\\expandafter{\\romannumeral4}, the accurate decoding of user communication data requires the knowledge of the environment, and if the data sent by the user is not recovered, the environmental information cannot be sensed accurately. Although a sufficient number of pilots can be used to implement the proposed environment sensing algorithm, an excessive number of pilots will cause a decrease in communication efficiency. To tackle this issue, we propose an iterative and incremental algorithm based on low-density pilots to jointly recover users' communication data and environmental information. Let ${{\\bf{s}}_k}\\left( {0 < k \\le K} \\right)$ denote the $k$-th data packets of all users. We insert a pilot $\\bf{P}$ before each $K$ data packets, and the AP can obtain the received data ${\\bf{y}}_{\\rm{p}}$. According to the previous section, based on the pilot $\\bf{P}$, the environmental information ${{\\bf{\\hat x}}_{\\rm{p}}}$ can be roughly estimated. Meanwhile, we need to use subsequent communication data packets to further improve the environment sensing results. Therefore, we use the pilot $\\bf{P}$, the received data ${\\bf{y}}_{\\rm{p}}$ and the estimated environmental information ${{\\bf{\\hat x}}_{\\rm{p}}}$ as initial terms to start the iterative algorithm.\n\n\\subsection{The Proposed Iterative Algorithm}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=6in]{fig4.eps}\n  \\caption{The proposed iterative algorithm.}\n  \\label{figIT}\n  \\end{figure*}\n\nAs shown in Fig. \\ref{figIT}, after receiving the $k$-th packet data ${\\bf{y}}_k$, the proposed iterative algorithm is divided into three parts:\n\n1. Forward propagation: First, use the estimated environment information ${{\\bf{\\hat x}}_{k - 1}}$ after receiving the $(k-1)$-th data packet to estimate the current channel ${{\\bf{\\hat H}}_k}$, and then the decoder decodes the received data ${\\bf{y}}_k$ to send data ${{\\bf{\\hat s}}_k}$ based on the estimation of current channel ${{\\bf{\\hat H}}_k}$. Finally, based on current ${\\bf{y}}_k$, ${{\\bf{\\hat s}}_k}$, the received data ${{\\bf{y}}_{k - {n_{\\rm{f}}} - 1}}, \\cdots ,{{\\bf{y}}_{k - 1}}$ and the decoded data ${{\\bf{\\hat s}}_{k - {n_{\\rm{f}}} - 1}}, \\cdots ,{{\\bf{\\hat s}}_{k - 1}}$ in the previous $n_{\\rm{f}}$ data packets, the current environmental information ${{\\bf{\\hat x}}_k}$ can be estimated more accurately. In the initial stage of the proposed algorithm, the results have not converged. Therefore, a certain number of iterations are required to make the system performance converge to a relatively accurate estimation of the transmitted data and environmental information, when $\\left\\lVert  {{{{\\bf{\\hat x}}}_k} - {{{\\bf{\\hat x}}}_{k - 1}}} \\right\\rVert _2 < {\\varepsilon _{\\rm{k}}}$, the ``momentum-mode'' is enabled.\n\n2. Self-iteration: First, estimate the current environment ${{\\bf{\\hat x}}_k}$ by using the channel estimated ${{\\bf{\\hat H}}_k}$ in the forward propagation part. Then, the decoder decodes the currently transmitted data ${{\\bf{\\hat s}}_k}$ again and estimates the current environment information ${{\\bf{\\hat x}}_k}$ again. \nFinally, iterate $K_{\\rm{s}}$ times to obtain more accurate transmission data and environmental information. $K_s$ can be set to a fixed value, or it can be gradually reduced as the algorithm converges, especially when $\\left\\lVert  {{{{\\bf{\\hat x}}}_k} - {{{\\bf{\\hat x}}}_{k - 1}}} \\right\\rVert_2 < {\\varepsilon _{\\rm{k}}}$, $K_s$ should be set to a small value. Enable the ``momentum-mode'' when $\\left\\lVert  {{{{\\bf{\\hat x}}}_k} - {{{\\bf{\\hat x}}}_{k - 1}}} \\right\\rVert_2 < {\\varepsilon _{\\rm{k}}}$.\n\n3. Feedback: feedback the estimated environmental information ${{\\bf{\\hat x}}_k}$ in the $k$-th data packet to the previous $n_{\\rm{b}}$ data packets, and based on the more accurate environmental information ${{\\bf{\\hat x}}_k}$ estimated by $k$-th data packet, estimate the received signal ${{\\bf{\\hat s}}_{k - {n_{\\rm{b}}} - 1}}, \\cdots ,{{\\bf{\\hat s}}_{k - 1}}$ again to improve the accuracy of decoding. Stop feedback when $\\left\\lVert {{{{\\bf{\\hat x}}}_k} - {{{\\bf{\\hat x}}}_{k - n_{\\rm{b}} - 1}}} \\right\\rVert_2 < {\\varepsilon _{\\rm{k}}}$.\n\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[width=6in]{fig15.eps}\n  \\caption{The Factor graph representation of the proposed iterative algorithm. }\n  \\label{figITGH}\n  \\end{figure*}\n\n\\begin{algorithm}[t]\n  \\caption{The proposed iterative algorithm}\n  \\label{alg2}\n  \\begin{algorithmic}[1] \n  \\REQUIRE\n  Calculated LOS channel matrix ${\\bf{H}}_r^{{\\rm{LOS}}}$, ${\\bf{H}}_r^{{\\rm{IRS1}}}$, ${\\bf{H}}_r^{{\\rm{s1}}}$, ${\\bf{H}}_r^{{\\rm{s2}}}$and ${\\bf{H}}_r^{{\\rm{s3}}}$. Given IRS reflection characteristic control matrix $\\bf{\\Theta}$.\n  \\STATE\n  \\textbf{Initialization}: Set pilot $\\bf{P}$. Defined sliding-window length $n_{\\rm{f}}$, $n_{\\rm{b}}$. Set ${\\varepsilon _{\\rm{k}}} > 0$, $0 < \\mu  < 1$, $K_{\\rm{s}} > 0$.\n  \\STATE\n  Estimate ${{\\bf{\\hat x}}_{\\rm{p}}}$ from $\\bf{P}$ and ${\\bf{y}}_{\\rm{p}}$ by GAMP. Let ${{\\bf{\\hat x}}_{\\rm{0}}} = {{\\bf{\\hat x}}_{\\rm{p}}}$, ${\\bf{y}}_0 = {\\bf{y}}_{\\rm{p}}$, ${{\\bf{\\hat s}}_0} = {\\bf{P}}$.\n  \\FOR {$k = 1, 2, \\cdots ,K$}\n\n \n  \\STATE\n  Estimate ${{\\bf{\\hat H}}_k}$ from ${{\\bf{\\hat x}}_{k - 1}}$ according to (\\ref{eq3}).\n  \\STATE\n  Estimate ${{\\bf{\\hat s}}_k}$ from ${\\bf{y}}_k$ and ${{\\bf{\\hat H}}_k}$ based on SCMA-IRS-MPA decoder.\n  \\STATE\n  Estimate ${{\\bf{\\hat x}}_k}$ from ${{\\bf{y}}_{k - {n_{\\rm{f}}} - 1}}, \\cdots ,{{\\bf{y}}_{k - 1}}$ and ${{\\bf{\\hat s}}_{k - {n_{\\rm{f}}} - 1}}, \\cdots ,{{\\bf{\\hat s}}_{k - 1}}$ according to formula (\\ref{eq31}).\n  \\STATE\n  Replace ${{\\bf{\\hat x}}_{k - 1}}$ with ${{\\bf{\\hat x}}_k}$, Repeat steps 3 to 5 $K_{\\rm{s}}$ times.\n \n  \\STATE\n  Estimate ${{\\bf{\\hat H}}_{k - 1}}, \\cdots ,{{\\bf{\\hat H}}_{k - {n_{\\rm{b}}} - 1}}$ from ${{\\bf{\\hat x}}_k}$ according to (\\ref{eq3}).\n  \\STATE\n  Estimate ${{\\bf{\\hat s}}_{k - 1}}, \\cdots ,{{\\bf{\\hat s}}_{k - {n_{\\rm{b}}} - 1}}$ from  ${{\\bf{y}}_{k - 1}}, \\cdots ,{{\\bf{y}}_{k - {n_{\\rm{b}}} - 1}}$ and ${{\\bf{\\hat H}}_{k - 1}}, \\cdots ,{{\\bf{\\hat H}}_{k - {n_{\\rm{b}}} - 1}}$ according to (\\ref{eq31}).\n  \\STATE\n  If $\\left\\lVert  {{{{\\bf{\\hat x}}}_k} - {{{\\bf{\\hat x}}}_{k - 1}}} \\right\\rVert_2 < {\\varepsilon _{\\rm{k}}}$, start ``momentum-mode'', else set $\\mu  = 0$.\n  \\ENDFOR\n  \\ENSURE\n  Estimated environment information ${{\\bf{\\hat x}}_k}$ and data packet ${{\\bf{\\hat s}}_1}, \\cdots ,{{\\bf{\\hat s}}_K}$.\n  \\end{algorithmic}\n  \\end{algorithm}\n\nWe summarize the proposed iterative algorithm in Algorithm \\ref{alg2}. During the execution process of the algorithm, the forward propagation process can be executed every time a new data packet is received. The self-iteration process can be executed as many times as necessary at any time, and the feedback process should be executed after the self-iteration process. Since the cached data cannot be too much, the forward propagation sliding-window size $n_{\\rm{f}}$ and the feedback window size $n_{\\rm{b}}$ need to be adjusted according to the actual system ability.\n\nThe effectiveness of the proposed algorithm can be explained by the message passing theory. Fig. \\ref{figITGH} shows the factor graph of the proposed iterative algorithm. The decoding algorithm and the sensing algorithm use each other's solution results as side information to achieve their performance. As the number of iterations increases, the environmental information and data information contained in the received data are separated and recovered. The components of the proposed iterative algorithm also reflect this idea: forward propagation passes the previously sensed environmental information and received data information to the next time slot so that the algorithm can incrementally optimize performance based on the continuously received data. The self-iteration process is executed repeatedly and iteratively based on the existing data, and the environmental information in the received data is fully obtained. The feedback process passes more accurate environmental information to the previous time slot and reduces the error caused by inaccurate decoding and sensing in the initial stage of the proposed algorithm.\n\n\\subsection{Computational Complexity Analysis}\nThe computational complexity of the proposed algorithm is mainly composed of two parts: \n\\begin{itemize}\n    \\item [(1)] \n    From the perspective of data decoding, we use the MPA decoder, whose computational complexity is $\\mathcal{O} \\left(Rd_{\\rm{f}}M^{d_{\\rm{f}}}+N_{\\rm{u}}M\\right)$. Compared with the ML decoder whose computational complexity is $\\mathcal{O}\\left(RC^{N_{\\rm{u}}}\\right)$, the computational complexity of the MPA decoder is lower. For example, when there are more users, the computational complexity of the MPA-based decoder just increases linearly. \n    \\item [(2)] \n    From the perspective of environment sensing, we use the GAMP algorithm, whose computational complexity is $\\mathcal{O}\\left(N_{\\rm{u}}N_{\\rm{R}}KN_{\\rm{s}}\\right)$. Compared with the OMP algorithm, whose computational complexity is $\\mathcal{O}(N_{\\rm{u}}N_{\\rm{R}}KN_{\\rm{s}} + (\\lambda N_{\\rm{s}})^3 )$, it can be seen that the GAMP algorithm is a relatively low-complexity CS reconstruction algorithm.\n   \n   \n\\end{itemize}\n\nIn summary, during the execution of the algorithm, replace $K$ with the sliding window sizes $n_{\\rm{f}}$ and $n_{\\rm{b}}$.\nThe computational complexity of the proposed iterative algorithm is $\\mathcal{O}( {R{d_{\\rm{f}}}{M^{{d_{\\rm{f}}}}}} $ $+ {N_{\\rm{u}}}M + {N_{\\rm{u}}}{N_{\\rm{R}}}{n_{\\rm{f}}}{n_{\\rm{b}}}{N_{\\rm{s}}} )$, where $n_{\\rm{f}}$ and $n_{\\rm{b}}$ can be controlled according to the convergence of the algorithm to save computing resources.\nIt can be seen that the computational complexity of the proposed algorithm is mainly determined by the number of users $N_{\\rm{u}}$ and the SCMA codebook parameter $d_{\\rm{f}}$. In contrast, if the OMP algorithm and the ML decoder are used to design the iterative algorithm, then its computational complexity will be $\\mathcal{O}( R{C^{{N_{\\rm{u}}}}} + {N_{\\rm{u}}}{N_{\\rm{R}}}{n_{\\rm{f}}}{n_{\\rm{b}}}{N_{\\rm{s}}} + (\\lambda N_{\\rm{s}})^3  )$, which is much higher than our algorithm.\n\nIn addition, the low complexity of the proposed iterative algorithm is also reflected in the use of low-density pilots, which effectively reduces the time-frequency resources and computing resources consumed by the pilots.\n\n\\section{System Performance Analysis}\nIn this section, we analyze the influence of the number of users on the decoding results of communication data and the accuracy of environment sensing. After receiving the $k$-th data packet, we calculate the mean square error (MSE) between the estimated environmental information ${{\\bf{\\hat x}}_k}$ and the actual environmental information $\\bf{x}$ to evaluate the accuracy of the environment sensing,\n\\begin{equation}\n{\\rm{MSE}} = \\frac{1}{{{N_{\\rm{s}}}}}\\left\\| {{{{\\bf{\\hat x}}}_k} - {\\bf{x}}} \\right\\|_2^2, \\label{eq33}\n\\end{equation}\nwhere $N_{\\rm{s}}$ is the total number of point clouds in the environment, and the SER between the decoding results ${{\\bf{\\hat s}}_k}$ and the original transmission data ${{\\bf{s}}_k}$ is calculated to evaluate the accuracy of data decoding.\n\nBased on the received data ${{\\bf{y}}_k}$ and decoded data ${{\\bf{\\hat s}}_k}$, the essence of estimating the environmental information ${{\\bf{\\hat x}}_k}$ is to solve the CS sparse reconstruction problem, and (16) can be expressed as\n\\begin{equation}\n{{{\\bf{\\hat x}}}_k} = \\arg \\mathop {\\min }\\limits_{\\bf{x}} {\\left\\| {\\bf{x}} \\right\\|_1}, \\label{eq34}\n\\end{equation}\n\\begin{equation}\n\\begin{array}{l}\n  {\\rm{s}}{\\rm{.t}}{\\rm{.}}\\quad {\\left\\| {{{\\bf{y}}_{k-{n_{\\rm{f}}},k}} - {{{\\bf{\\hat s}}}_{k-{n_{\\rm{f}}},k}}{{\\bf{x}}^{\\rm{T}}} {{\\bf{\\tilde{A}}}^{\\rm{s}}\\left(\\left[k-n_{\\rm{f}},k\\right]\\right)} } \\right\\|_2}\\\\\n  \\quad  {\\left\\| { {\\left( {{{\\bf{s}}_{k-{n_{\\rm{f}}},k}} - {{{\\bf{\\hat s}}}_{k-{n_{\\rm{f}}},k}}} \\right){{\\bf{x}}^{\\rm{T}}}{{\\bf{\\tilde{A}}}^{\\rm{s}}\\left(\\left[k-n_{\\rm{f}},k\\right]\\right)}} } \\right\\|_2} + {\\varepsilon _{\\rm{x}}}\n  \\end{array}, \\label{eq35}\n\\end{equation}\nwhere ${{\\bf{y}}_{k-{n_{\\rm{f}}},k}}$, ${{\\bf{\\hat s}}_{k-{n_{\\rm{f}}},k}}$, and ${{\\bf{\\tilde{A}}}^{\\rm{s}}\\left(\\left[k-n_{\\rm{f}},k\\right]\\right)}$ represent the received packets, the decoding results, and the measurement matrix in the forward propagation window of size $n_{\\rm{f}}$ respectively, ${\\rm{SER}} \\propto {\\left\\| { {\\left( {{{\\bf{s}}_{k-{n_{\\rm{f}}},k}} - {{{\\bf{\\hat s}}}_{k-{n_{\\rm{f}}},k}}} \\right){{\\bf{x}}^{\\rm{T}}}{{\\bf{\\tilde{A}}}^{\\rm{s}}\\left(\\left[k-n_{\\rm{f}},k\\right]\\right)}} } \\right\\|_2}$. Therefore, when the decoding error rate (SER) increases, the constraint conditions of the sparse reconstruction problem become more slack, and the estimated environmental information error (MSE) increased. According to the theory of CS \\cite{Donoho}, the theoretical upper bound of environment sensing accuracy is,\n\\begin{equation}\n{\\left\\| {{\\bf{x}} - {{{\\bf{\\hat x}}}_k}} \\right\\|_2} \\le c \\cdot {R_p} \\cdot {\\left( {\\frac{{{N_{\\rm{u}}} \\cdot {N_{\\rm{R}}} \\cdot {n_{\\rm{f}}}}}{{\\log {N_{\\rm{s}}}}}} \\right)^{{1 \\mathord{\\left\/\n {\\vphantom {1 2}} \\right.\n \\kern-\\nulldelimiterspace} 2} - {1 \\mathord{\\left\/\n {\\vphantom {1 p}} \\right.\n \\kern-\\nulldelimiterspace} p}}}, \\label{eq36}\n\\end{equation}\nwhere $c > 0$ is a constant, ${\\left\\| {\\bf{x}} \\right\\|_p} \\le {R_p},\\left( {0 < p < 2} \\right)$ is the sparsity condition, ${\\rm{MSE}} \\propto {\\left\\| {{\\bf{x}} - {{{\\bf{\\hat x}}}_k}} \\right\\|_2}$. Therefore, when the number of users $N_{\\rm{u}}$ increases, the rank of the measurement matrix in the sparse reconstruction problem increases, and the environment sensing error (MSE) decreases. In addition, when the number of non-zero elements in the environment increases, $R_p$ increases, and the environment sensing error (MSE) increases.\n\nWhen using MPA decoder to decode the received signal based on the ML theory, for a single ORE, let $C = N_{\\rm{u}} \\times M$ be the number of symbols in the codebook, then $N_{\\rm{u}}$ users at each moment have ${C^{{N_{\\rm{u}}}}}$ ways to send symbols. The theoretical upper bound of the average decoding error rate (SER) is,\n\\begin{equation}\n{\\rm{SER}} \\le \\frac{1}{{{C^{{N_{\\rm{u}}}}}}}\\sum\\limits_{{{\\bf{s}}_{\\rm{a}}}} {\\sum\\limits_{{{\\bf{s}}_{\\rm{b}}},{{\\bf{s}}_{\\rm{a}}} \\ne {{\\bf{s}}_{\\rm{b}}}} {P\\left( {{{\\bf{s}}_{\\rm{a}}} \\to {{\\bf{s}}_{\\rm{b}}}} \\right)} }, \\label{eq37}\n\\end{equation}\nwhere $P\\left( {{{\\bf{s}}_{\\rm{a}}} \\to {{\\bf{s}}_{\\rm{b}}}} \\right)$ represents the pairwise error probability (PEP) that the symbol ${\\bf{s}}_{\\rm{a}}$ is incorrectly decoded to ${\\bf{s}}_{\\rm{b}}$. In (\\ref{eq37}), there are a total of $\\left( {C - 1} \\right){C^{2{N_{\\rm{u}}} - 1}}$ items for summation. Therefore, when other conditions are the same, SER increases as the number of users $N_{\\rm{u}}$ increases. Due to the random distribution of scatterers in the environment, the channels from the users to the AP can be expressed as Rayleigh fading channels. According to the ML theory, the decoding process in (\\ref{eq8}) can be expressed as\n\\begin{equation}\n  {{{\\bf{\\hat s}}}_k} = \\arg \\mathop {\\min }\\limits_{j \\in {C^{{n_{\\rm{u}}}}}}{\\left\\| {{{\\bf{y}}_k} - {{{\\bf{\\hat H}}_k}{{\\bf{s}}_k}\\left( j \\right)} } \\right\\|^2}, \\label{eq38}\n\\end{equation}\n\\begin{equation}\n  {\\rm{s}}{\\rm{.t}}{\\rm{.}}\\quad {{\\bf{y}}_k} = {{{\\bf{\\hat H}}}_k}{{\\bf{s}}_k} + \\left( {{{\\bf{H}}_k} - {{{\\bf{\\hat H}}}_k}} \\right){{\\bf{s}}_k} + {\\bf{w}}, \\label{eq39}\n\\end{equation}\nwhere $\\bf{w}$ represents Gaussian white noise with variance $N_0$. Due to the random distribution of scatterers in the environment, the interference caused by channel estimation errors is also Gaussian. Then the PEP in (\\ref{eq37}) can be expressed as\n\\begin{equation}\nP\\left( {{{\\bf{s}}_{\\rm{a}}} \\to {{\\bf{s}}_{\\rm{b}}}} \\right) = {{\\mathbb{E}}_{{{{\\bf{\\hat H}}}_k}}}\\left[ {Q\\left( {\\sqrt {\\frac{{{{\\left\\| {{{{\\bf{\\hat H}}}_k}\\left( {{{\\bf{s}}_{\\rm{a}}} - {{\\bf{s}}_{\\rm{b}}}} \\right)} \\right\\|}^2}}}{{{N_0} + \\mathbb{D} \\left( {\\left( {{{\\bf{H}}_k} - {{{\\bf{\\hat H}}}_k}} \\right){{\\bf{s}}_k}} \\right)}}} } \\right)} \\right], \\label{eq40}\n\\end{equation}\nWhere $Q$ is the error function. Therefore, the channel estimation error caused by inaccurate environmental information estimation will lead to an increase in the decoding symbol error rate (SER).\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=7.5cm]{fig5.eps}\n  \\caption{The trade-off relationship between the number of users and system performance.}\n  \\label{figtrade}\n  \\end{figure}\n\nAs shown in Fig. \\ref{figtrade}, the solid line indicates that the increase in the number of users $N_{\\rm{u}}$ promotes the accuracy of environment sensing, and the decrease of MSE reduces the channel prior information error $\\Delta {{\\bf{\\hat H}}_k}$ for decoding, so the SER also decreases and the decoding becomes more accurate. On the other hand, the dotted line indicates that an increase in the number of users $N_{\\rm{u}}$ increases the error of decoding (SER), and the decoded data error $\\Delta {{\\bf{\\hat s}}_k}$ also increases, which increases the environment sensing error (MSE). Therefore, firstly, when the number of users decreases, the environment sensing error (MSE) increases. Secondly, when the number of users increases, the SCMA decoding error (SER) increases. Finally, because the proposed iterative algorithm repeatedly executes environment sensing and data decoding, their performance affects each other, resulting in the same trend of SER and MSE, the number of users should be a compromise choice. The number of users $N_{\\rm{u}}$ has a trade-off relationship with system performance (MSE\/SER) because the number of users $N_{\\rm{u}}$ is traded for system performance. Finally, the optimal operating point could be estimated as\n\\begin{equation}\n{\\tilde N_{\\rm{u}}} = \\arg \\mathop {\\min }\\limits_{{N_{\\rm{u}}}} \\quad {a_1} \\cdot {\\rm{MSE}} + {a_2} \\cdot {\\rm{SER}}. \\label{eq41}\n\\end{equation}\n\nWhen the number of users is ${\\tilde N_{\\rm{u}}}$, the system performance MSE and SER reach the best. In practical applications, we need to adjust the coefficients $a_1$ and $a_2$ to suit the system's requirements for communication and environment sensing performance. And select the appropriate SCMA codebook and system parameters according to the number of users in the scene to ensure that the number of actual users is within the optimal working range of the system.\n\n\\section{Numerical Results}\nIn this section, we simulated the performance of the algorithm and all simulations are conducted in MATLAB 2017b on a computing server with a Xeon E5-2697 v3 processer and 128GB memory. The simulation scenario is set in a room with a size of $4\\rm{m} \\times 4m \\times 4m$, and the point cloud with a size of $8 \\times 8 \\times 8$ is used to represent the environmental information. The transmission signal frequency is set to 28 - 30GHz, and the bandwidth is 2GHz. A $20 \\times 20$ IRS is used for assist communication. In order to meet the actual system design, we set the IRS amplitude reflection coefficient $\\rho_{{n_{\\rm{I}}}}  = 1$, the phase shift $\\varphi_{{n_{\\rm{I}}}}   = 0$ or $\\pi $. The position of the scatterers distributed in the space is random, and the scattering coefficient ${\\bf{x}}_{{n_{\\rm{s}}}} \\in \\left[ {0,1} \\right]$. As shown in Fig. \\ref{figanswer}, small cubes are used to indicate the position distribution and scattering coefficient of the point cloud in space. The lower the transparency of the small cube, the larger the scattering coefficient of the point.\n\\begin{figure*}[t]\n  \\centering\n  \\subfigure[The original environment scatterer distribution.]{\n  \\includegraphics[width=0.4\\textwidth]{fig6.eps}}\n  \\subfigure[The sensing result when $\\rm{E_b\/N_0}=0dB$.]{\n  \\includegraphics[width=0.4\\textwidth]{fig7.eps}}\n \\subfigure[The sensing result when $\\rm{E_b\/N_0}=5dB$.]{\n  \\includegraphics[width=0.4\\textwidth]{fig8.eps}}\n  \\subfigure[The sensing result when $\\rm{E_b\/N_0}=10dB$.]{\n  \\includegraphics[width=0.4\\textwidth]{fig9.eps}}\n  \\caption{The original environment scatterer distribution and the sensing results under different SNR conditions.}\n  \\label{figanswer}\n  \\end{figure*}\n\nThe distribution of the environment scatterer is shown in Fig. \\ref{figanswer}(a), and the system parameters are set to the number of users $N_{\\rm{u}}$ = 6, the number of OREs $R$ = 4, and $d_{\\rm{v}}$ = 2. According to the convergence of the algorithm, we set $K_{\\rm{s}} = 5$ in this section. After the proposed algorithm is iterated to convergence, the intuitive sensing results are shown in Fig. \\ref{figanswer}(b), Fig. \\ref{figanswer}(c), and Fig. \\ref{figanswer}(d), when the signal-to-noise ratio (SNR, $\\rm{E_b\/N_0}$) are 0dB, 5dB, and 10dB respectively. It can be seen that the sensing result is very blurred when $\\rm{E_b\/N_0}=0dB$, and the shape of the target object can only be barely distinguished. As the SNR condition becomes better, the number of misidentified scatterers gradually decreases until $\\rm{E_b\/N_0}=10dB$ can clearly distinguish the shape of the target. This is because the SNR conditions affect the accuracy of communication data decoding, and therefore affect the accuracy of the sensing results.\n\nAs shown in Fig. \\ref{IT-MSE} and Fig. \\ref{IT-SER}, the system parameters are set to the number of users $N_{\\rm{u}}$ = 6, the number of OREs $R$ = 4, $d_{\\rm{v}}$ = 2, and the sparsity of randomly generated environmental scatterers is 1.5\\%. Set the forward propagation window size $n_{\\rm{f}}$ = 10 and the feedback window size $n_{\\rm{b}}$ = 1. We use MSE to evaluate the environmental sensing accuracy, when the number of data packets increases, the iterative algorithm converged, and the environment sensing result gradually becomes accurate. We use SER to evaluate the accuracy of transmission data decoding. As the number of data packets increases, the iterative algorithm converged, and the transmission data decoding results become more accurate. \nAt the same time, due to the existence of feedback, as the number of data packets increases, the data packets at the previous time are decoded based on the more accurate environmental information at the later time. The feedback process improves the decoding accuracy during convergence.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8cm]{fig17.eps}\n  \\caption{The relationship between the data packet number (number of iterations) and MSE.}\n  \\label{IT-MSE}\n  \\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8cm]{fig16.eps}\n  \\caption{The relationship between the data packet number (number of iterations) and SER.}\n  \\label{IT-SER}\n  \\end{figure}\n\nAfter the proposed algorithm iterates to convergence, the relationship between the number of users and system performance is shown in Fig. \\ref{UE-MSE} and Fig. \\ref{UE-SER}. We set a tough condition and the system parameters are set to the number of OREs $R$ = 7, $d_{\\rm{v}}$ = 2, $\\rm{E_b\/N_0}=10dB$, and the sparsity of randomly generated environmental scatterers is 3\\%. \nAs analyzed in the section \\uppercase\\expandafter{\\romannumeral6}, there is a trade-off relationship between the number of users and the system performance indicators SER and MSE. \nAs the number of users changes, the environment sensing performance and the multi-user communication performance cannot reach the best at the same time.\nAs shown in Fig. \\ref{UE-MSE} and Fig. \\ref{UE-SER}, \nwhen the number of users is small, as the number of users increases, the sensing accuracy is significantly improved (as analyzed in \\eqref{eq36}), and therefore the decoding accuracy is improved, until the optimal operating point ${\\tilde N_{\\rm{u}}} = 12$ is reached under simulation conditions. When the sensing of the environment is accurate sufficiently, a further increase in the number of users causes a decrease in the accuracy of decoding (as analyzed in \\eqref{eq37} and \\eqref{eq40}), and therefore the accuracy of environment sensing also decreases slightly.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8cm]{fig12.eps}\n  \\caption{The trade-off relationship between the number of users and MSE.}\n  \\label{UE-MSE}\n  \\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8cm]{fig13.eps}\n  \\caption{The trade-off relationship between the number of users and SER.}\n  \\label{UE-SER}\n  \\end{figure}\n\nFig. \\ref{UE-MSE} and Fig. \\ref{UE-SER} also show the impact of ``momentum-mode'' on system performance. The ``momentum-mode'' uses data information outside the sliding-window for environmental sensing. As analyzed in Section \\uppercase\\expandafter{\\romannumeral6}, the environment sensing accuracy is low when the number of users is small, so the ``momentum-mode'' accumulates errors and cannot improve system performance. However, ``momentum-mode'' can improve system performance when the number of users is higher than the optimal operating point, Since. It can be seen from the simulation results that when $\\mu $ = 0.9, the momentum coefficient is large, the MSE and SER are greatly improved when there is a large number of users, and there is a negative impact on the SER in the case of few users. When $\\mu $ = 0.5, the momentum coefficient is moderate, the MSE and SER indicators are slightly improved when there is a large number of users, and there is no significant impact on the system performance in the case of few users. When $\\mu $ = 0.1, the momentum coefficient is small, there is no significant impact on the system performance. We recommend using a larger momentum coefficient when there is a large number of users, and using a moderate or small momentum coefficient in the case of few users.\n\nFinally, in Table \\ref{tab1}, we provide the run time of using the $k$-th received data package for data decoding and environment sensing (steps 4 to 10 in Algorithm \\ref{alg2}). The simulation parameter settings are the same as those in Fig. \\ref{UE-MSE} and Fig. \\ref{UE-SER} and the momentum mode is not enabled. As shown in Table \\ref{tab1}, we verify that the computational complexity of the algorithm increases with the increase in the number of users.\n\\begin{table}[h]\n  \\centering\n  \\caption{The run time of proposed algorithm.}\n  \\begin{tabular}{|p{1in}|p{0.3in}|p{0.3in}|p{0.3in}|p{0.3in}|}\n      \\arrayrulecolor{black}\n      \\hline  \n      \\makecell[l]{Number of users $N_{\\rm{u}}$} & \\makecell[c]{5} & \\makecell[c]{10} & \\makecell[c]{15} & \\makecell[c]{20}\\\\\n      \\hline \n      \\makecell[l]{Run time (s)} & \\makecell[c]{4.95} & \\makecell[c]{5.11} & \\makecell[c]{5.33} & \\makecell[c]{5.64}\\\\\n      \\hline \n      \\end{tabular}\n      \\label{tab1}\n  \\end{table}\n\n\\section{Conclusion}\nIn the diverse wireless communication application scenarios in the future, environment sensing is an important component of the wireless communication system. In the scenario of IRS-assisted indoor uplink communication, we design a multiple access method and an environment sensing method. The multiple access method is based on SCMA. With the assistance of the IRS, based on the sparse codebook of the transmitted signals, the SCMA-IRS-MPA decoder is used. The environment sensing algorithm is based on the CS theory, including time sliding-window and ``momentum-mode'' which keep on sensing the environment while continuously receiving the data stream sent by the user. In this paper, the proposed multiple access algorithm and the proposed environment sensing algorithm rely on each other. Therefore, we propose a novel iterative algorithm based on low-density pilots to jointly solve the multiple access and environment sensing problems and achieve the integration of environment sensing and communication. Finally, numerical simulation has verified the convergence and effectiveness of the iterative and incremental algorithm and analyzed the trade-off relationship between the number of users and system performance. We also give a system parameter configuration method. The sensing-communication integration ideas and algorithms proposed in this paper will provide references for the development of new wireless communication technologies in the future.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}