diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzcoxh" "b/data_all_eng_slimpj/shuffled/split2/finalzzcoxh" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzcoxh" @@ -0,0 +1,5 @@ +{"text":"\\section{$\\text{Introduction}$}\n\n General Relativity (GR) as the standard model of gravity theory is tested well, especially in the solar system. But it also confronts some unanswered questions, such as the problems on the dark matter \\cite{DM1,DM2}, the late accelerating universe \\cite{SN-acc1,SN-acc2}, the inflation \\cite{inflation1,inflation2}, and the quantization, etc. Maybe, the gravity does not work in the framework of the GR theory at the large scales and the high-energy region. In order to explore the \"ultimate\" theory that adequately describe gravitational interaction, modified or extended gravity theories have been widely investigated \\cite{mg1,mg2,mg3,mg4,mg5,mg6,mg7,mg8,mg9,mg10}, such as the $f(R)$ theory \\cite{fr-review1,fr-review2,fr-3,fr-4}, the $f(T)$ theory \\cite{ft1,ft2}, the $f(G)$ theory \\cite{fg1,fg2}, the Brans-Dicke (BD) theory \\cite{original-BD}, and so on.\n\n\n In the BD theory, the Newton gravity constant $G(t)=1\/\\phi(t)$ is considered as a function of time. Thus, a scalar field $\\phi$ can be introduced naturally into the action. Some observational constraints on $G(t)$ can be found in Refs. \\cite{VG-MNRAS-2004-dwarf,VG-PRD-2004-white,VG-APJ,VG-PRD-2002-SN,VG-PRL-1996-neutron,vg-prl-constraint,vg-constraint-lu-prd,vg-constraint-lu-epjc}. The action of the BD theory has a form \\cite{bd-action}\n \\begin{equation}\n S_{BD}=\\int \\sqrt{-g}d^{4}x[\\frac{1}{16\\pi}(\\phi R-\\frac{\\omega}{\\phi}\\partial_{\\mu}\\phi\\partial^{\\mu}\\phi)+L_{m}],\\label{action-BD}\n \\end{equation}\n where $\\omega$ is the coupling constant. For observational and theoretical motivations, several extended versions of the BD theory have been developed, such as adding a potential term to the original BD theory \\cite{BD-potential}, assuming the coupling constant $\\omega$ to be variable with respect to time \\cite{BD-omegat1,BD-omegat2}, generalizing $\\phi$ to be a function $f(\\phi)$ in the coupling term $\\phi R$ \\cite{BD-ST,BD-ST1}, choosing a higher-dimension geometry in the BD theory \\cite{Ma-5DBD}, etc. The applications of these extended BD theories have been investigated widely, such as at the aspects of cosmology \\cite{GBD-cosmic1,GBD-cosmic2,GBD-cosmic3}, weak-field approximation \\cite{GBD-weak}, observational constraints \\cite{GBD-constraint1,GBD-constraint2}, and so on \\cite{BD-widely1,BD-widely2,BD-widely3}.\n\n\n Recently, a different strategy was proposed to modify the BD theory (called GBD) in Refs. \\cite{GBD-L,GBD} by generalizing the Ricci scalar $R$ to be an arbitrary function $f(R)$ in the BD action. Comparing with other modified theories, one can find that the GBD theory in the metric formalism have some interesting properties or solve some problems existing in other theories \\cite{GBD-L,GBD}. Several results could be exhibited briefly as follows. (1) The state parameter of geometrical dark energy in the GBD model can cross over the phantom boundary $w=-1$ as achieved in the quintom model, without bearing the problems existing in the quintom model \\cite{GBD} (in the double scalar-fields quintom model, it is required to include both the canonical quintessence field and the non-canonical phantom field in order to make the state parameter to cross over $w=-1$ \\cite{quintom,quintom1}, while several fundamental problems are associated with phantom field, such as the problem of negative kinetic term and the fine-tuning problem, etc). (2) One knows that the metric-formalism $f(R)$ theories are equivalent to the BD theory with a potential (abbreviated as BDV) for taking a specific value of the coupling parameter $\\omega=0$ \\cite{fr-review2}, where the specific choice: $\\omega=0$ is quite exceptional, and it is hard to understand the corresponding absence of the kinetic-energy term for the field in the action. However, the BD field in the GBD theory owns the non-disappeared kinetic term in the action \\cite{GBD-L,GBD}. (3) Using the method of the weak-field approximation Ref. \\cite{GBD-L} showed that the GBD theory could solve the problem of $\\gamma$ value emerging in $f(R)$ modified gravity (i.e. the inconsistent problem between the observational $\\gamma$ value and the theoretical $\\gamma$ value \\cite{fr-review2,ppn-fr1,ppn-fr2}), without introducing the so-called chameleon mechanism. Here $\\gamma$ is the parametrized post-Newtonian (PPN) parameter. Furthermore, the GBD theory tends to investigate the physics from the viewpoint of geometry, while the BDV or the two scalar-fields quintom model tends to solve physical problems from the viewpoint of matter. It is possible that several special characteristics of scalar field could be revealed through studies of the gravitational geometry in the GBD theory.\n\n\n It is well known that some assumptions have to be taken at prior for developing modified gravity theories of GR. For example: (1) which one or ones of the dynamical variables (the metric, the connection, or the tetrad, etc.) should be chosen to describe the gravitational interaction? (2) As a covariant action theory, what is the constructed Lagrangian form for the gravitational system, and which corresponding space-time geometry (the Riemann, the Weitzenb$\\ddot{o}$ck, the Riemann-Cartan, or the higher-dimension geometry, etc.) should be used? Then based on the constructed Lagrangian quantity, one can derive the field equations by using the variational principle. And the surviving theories should pass through the tests of experiments from the solar system, the astrophysics, the cosmology, etc.\n\n\n As shown in some references, the gravity theories defining in the Riemann geometry are often depicted in two formalisms: the metric formalism \\cite{fr-review2,metric-variable} and the Palatini formalism \\cite{Pala-variable,Pala-variable1}. In the metric formalism (or the standard formalism) the Levi-Civta connection is related to metric, while in the Palatini formalism the metric and the connection are considered as the independent dynamical variables. For a general non-linear $f(R)$ function, the different field equations can be obtained for the metric-$f(R)$ theories and the Palatini-$f(\\tilde{R})$ theories \\cite{fr-review2}, respectively. Comparing to the metric formalism, some advantages of the Palatini approach could be found. For example: (1) the field equations in the metric formalism are the fourth-order PDE (partial differential equation), while the field equations in the Palatini formalism are the second-order PDE which is easier to solve and interpret \\cite{Pala-merit1}; (2) Palatini-formalism theory raises the effective cutoff of the theory without introducing additional degrees of freedom below the Planck scale \\cite{Pala-merit2}; (3) Palatini-formalism theory could lead to different inflationary predictions, which opens a door to test the nature of gravity by using the future cosmological observations \\cite{Pala-merit3}; (4) Palatini-formalism theory could lead to different interactions among the Standard Model particles and the Higgs field in the large field regime, with a potential impact on the entropy production process following the end of inflation \\cite{Pala-merit4}, etc. \\cite{Pala-merit5,Pala-merit6}.\n\n\n\n In this paper, we investigate the GBD theory in the framework of Palatini formalism. The field equations and the linearized equations are derived in the Palatini-GBD theory. Comparing to the metric formalism of GBD theory \\cite{GBD-L}, some new results can be found in the Palatini-formalism of GBD theory. (1) The geometrical scalar field in the Palatini-formalism of GBD theory is massless and source-free. These properties of geometric scalar field are different from the results given in the metric-formalism of GBD theory \\cite{GBD-L}; (2) The parameterized post Newton parameter in the Palatini-GBD theory has a form: $\\gamma=\\frac{2\\omega-F_{0}}{2\\omega+F_{0}}$, which could pass through the experimental test: $ |\\omega| > 40000$. Comparing to the Ref. \\cite{GBD-L}, we can see a different expression of $\\gamma$ in the metric-GBD theory, where $\\gamma$ depends on the parameters: $\\omega$, $F_{0}$ and $m_{s}$. (3) Given that investigating the polarization of gravitational waves (GWs) can serve to discriminate the different gravitational theories, we also study the polarization modes of GWs in the Palatini-GBD theory. The Newman-Penrose (NP) method \\cite{gw-polar4,gw-polar5} and the geodesic deviation (GD) method \\cite{gw-polar6} are used to explore the polarization modes of GWs in the GBD theory. It is observed that the polarization modes of GWs in the Palatini-GBD theory (the three polarization types) are different from that in the metric-GBD theory (the four polarization types). The results in the GBD theory also show that the extra scalar field $F$ and the BD scalar field generate the new polarizations (the scalar polarization modes) for GWs which are not present in the standard GR or the Palatini-$f(R)$ theory.\n\n\n The structure of our paper are as follows. In section II, we derive the basic equations in the Palatini-GBD theory. In section III, the linearized field equations are gained by using the method of the weak-field approximation. Section IV investigate the polarization modes of gravitational waves in the Palatini-GBD theory. Section V discuss the parameterized post Newton parameter (PPN). Section VI is the conclusions.\n\n\n\n\n\\section{$\\text{Field equations in the Palatini-formalism of GBD theory}$}\n\n\nThis section is devoted to derive the basic equations in the Palatini-GBD theory. The action of the GBD theory in the Palatini formalism read as,\n\\begin{equation}\nS=S_g(g_{\\mu \\nu },\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu},\\phi)+S_{\\phi}(g_{\\mu \\nu },\\phi)+S_m(g_{\\mu \\nu },\\psi )=\\frac{1}{2}\\int d^4x{\\cal L}_{T},\\label{action}\n\\end{equation}\nwith the total Lagrange quantity\n\\begin{equation}\n{\\cal L}_T=\\sqrt{-g}[\\phi f(\\tilde{R})- \\frac{\\omega}{\\phi}\\partial _\\mu \\phi \\partial ^\\mu \\phi+\\frac{16\\pi }{c^4}{\\cal L}_m].\\label{lagrange}\n\\end{equation}\nHere the metric $g_{\\mu \\nu }$ and the connection $\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}$ are considered as the independent dynamical variables, $g$ denotes the determinant of $g_{\\mu\\nu}$, and ${\\cal L}_m$ denotes the matter Lagrangian associated with the matter field $\\psi$ and $g_{\\mu\\nu}$. $f(\\tilde{R})$ is an arbitrary function of Ricci scalar: $\\tilde{R}=g_{\\mu\\nu}\\tilde{R}^{\\mu\\nu}$, and the Ricci tensor $\\tilde{R}_{\\mu\\nu}$ is defined by the independent Palatini connection $\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}$\n\\begin{equation}\n\\tilde{R}_{\\mu\\nu}=\\tilde{R}^{\\alpha}_{\\mu\\alpha\\nu}=\\partial_{\\lambda}\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}-\\partial_{\\mu}\\widetilde{\\Gamma}^{\\lambda}_{\\lambda\\nu}\n+\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}\\widetilde{\\Gamma}^{\\rho}_{\\rho\\lambda}-\\widetilde{\\Gamma}^{\\lambda}_{\\nu\\rho}\\widetilde{\\Gamma}^{\\rho}_{\\mu\\lambda}\n\\label{Ricci-Tensor-Pala}.\n\\end{equation}\n Using the variational principle, we can derive the evolutional equations of the dynamical fields in the Palatini-formalism of GBD theory. Varying the action (\\ref{action}) with respect to $g_{\\mu\\nu}$ and $\\phi$, we gain two field equations as follows\n\\begin{eqnarray}\n\\phi \\left[ F(\\tilde{R})\\tilde{R}_{\\mu \\nu }-\\frac{1}{2}f(\\tilde{R})g_{\\mu \\nu }\\right]+ \\frac{1}{2}\\frac{\\omega}{\\phi}g_{\\mu\\nu}\\partial_\\sigma\\phi\\partial^\\sigma\\phi\n-\\frac{\\omega}{\\phi}\\partial_\\mu\\phi\\partial_\\nu\\phi = 8\\pi T_{\\mu \\nu },\\label{gravitational-eq-Pala}\n\\end{eqnarray}\n\\begin{equation}\nf(\\tilde{R})+2\\omega\\frac{\\Box \\phi}{\\phi} -\\frac{\\omega}{\\phi^{2}}\\partial _\\mu \\phi \\partial ^\\mu \\phi=0,\\label{BD-scalar-eq}\n\\end{equation}\nwhere $F(\\tilde{R})\\equiv\\partial f(\\tilde{R})\/\\partial \\tilde{R}$, $\\Box \\equiv \\nabla ^\\mu \\nabla _\\mu $ and $T_{\\mu \\nu }=\\frac{-2}{\\sqrt{-g}}\\frac{\\delta S_m}{\\delta g^{\\mu \\nu }}$ is the energy-momentum tensor of matter. The trace of Eq.(\\ref{gravitational-eq-Pala}) is\n\\begin{eqnarray}\nF(\\tilde{R})\\tilde{R}-2f(\\tilde{R})+\\frac{\\omega}{\\phi^{2}}\\partial_\\mu\\phi\\partial^\\mu\\phi = \\frac{8\\pi T}{\\phi}.\\label{trace-grav-Pala}\n\\end{eqnarray}\nVarying the action with respect to $\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}$ gives\n\\begin{eqnarray}\n\\widetilde{\\nabla}_{\\lambda}(\\sqrt{-g}\\phi F(\\tilde{R})g^{\\mu\\nu})=0,\\label{connection-eq}\n\\end{eqnarray}\nwhere $\\widetilde{\\nabla}$ is the covariant derivative with respect to the Palatini connection. Eq.(\\ref{connection-eq}) implies that the connection can be represented as the Christoffel symbol associated with the metric $h_{\\mu\\nu}$ by defining $h_{\\mu\\nu}=\\phi F(\\tilde{R})g_{\\mu\\nu}$. Then we can have a following relation\n\\begin{equation}\n\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}=\\Gamma^{\\lambda}_{\\mu\\nu}+\\frac{1}{2\\phi F}[-g_{\\mu\\nu}\\partial^{\\lambda}(\\phi F)+\\delta^{\\lambda}_{\\nu}\\partial_{\\mu}(\\phi F)+\\delta^{\\lambda}_{\\mu}\\partial_{\\nu}(\\phi F)],\\label{connection-relation}\n\\end{equation}\nwhere $\\Gamma^{\\lambda}_{\\mu\\nu}$ is the Livi-Civita connection associated with the metric $g_{\\mu\\nu}$. Thus, by using Eq.(\\ref{Ricci-Tensor-Pala}) the Ricci tensor and the Ricci scalar in the Palatini formalism are rewritten as\n\\begin{equation}\n\\tilde{R}_{\\mu\\nu}=R_{(g)\\mu\\nu}+\\frac{3}{2(\\phi F)^{2}}\\nabla_{\\mu}(\\phi F)\\nabla_{\\nu}(\\phi F)-\\frac{1}{\\phi F}\\nabla_{\\mu}\\nabla_{\\nu}(\\phi F)-\\frac{1}{2\\phi F}g_{\\mu\\nu}\\Box(\\phi F),\\label{Ricci-Tensor-metric}\n\\end{equation}\n\\begin{equation}\n\\tilde{R}=R_{(g)}+\\frac{3}{2(\\phi F)^{2}}\\nabla^{\\sigma}(\\phi F)\\nabla_{\\sigma}(\\phi F)-\\frac{3}{\\phi F}\\Box(\\phi F),\\label{Ricci-scalar-metric}\n\\end{equation}\nwhere $R_{(g)\\mu\\nu}$ and $R_{(g)}$ denotes the Ricci tensor and the Ricci scalar defining in the metric formalism, and all covariant derivatives are taken with respect to the metric $g_{\\mu\\nu}$. Combining above equations, the modified Einstein equation is derived as\n\\begin{equation}\nG_{\\mu\\nu}=R_{(g)\\mu\\nu}-\\frac{1}{2}R_{(g)}g_{\\mu\\nu}=\\frac{8\\pi T_{\\mu\\nu}}{\\phi F}+8\\pi T_{\\mu\\nu}^{eff}\\label{Einstain-Tensor-Pala}\n\\end{equation}\nwith\n\\begin{equation}\n\\begin{split}\n8\\pi T_{\\mu\\nu}^{eff}=-\\frac{\\omega}{2\\phi^2F}g_{\\mu\\nu}\\partial_{\\sigma}\\phi\\partial^{\\sigma}\\phi+\\frac{f}{2F}g_{\\mu\\nu}+\\frac{\\omega}{\\phi^{2} F}\\partial_{\\mu}\\phi\\partial_{\\nu}\\phi-\\frac{3}{2(\\phi F)^{2}}\\nabla_{\\mu}(\\phi F)\\nabla_{\\nu}(\\phi F) \\\\\n+\\frac{1}{\\phi F}\\nabla_{\\mu}\\nabla_{\\nu}(\\phi F)-\\frac{1}{2}g_{\\mu\\nu}\\tilde{R}+\\frac{3}{4(\\phi F)^{2}}g_{\\mu\\nu}\\nabla^{\\sigma}(\\phi F)\\nabla_{\\sigma}(\\phi F)-\\frac{1}{\\phi F}g_{\\mu\\nu}\\Box(\\phi F).\\label{re-gravitation-eq}\n\\end{split}\n\\end{equation}\nWhen $f(\\tilde{R})$ linear in $\\tilde{R}$, Eq.(\\ref{Einstain-Tensor-Pala}) is identical to the field equation in the metric formalism of the GBD theory \\cite{GBD}. The trace of the gravitational field equation and the BD field equation have the forms:\n\\begin{equation}\n\\Box (\\phi F)=\\frac{8\\pi T}{3}-\\frac{\\omega}{3\\phi}\\partial_{\\mu}\\phi\\partial^{\\mu}\\phi+\\frac{2\\phi f(\\tilde{R})}{3}-\\frac{2\\phi F}{3}\\tilde{R}+\\frac{1}{2\\phi F}\\nabla_{\\mu}(\\phi F)\\nabla^{\\mu}(\\phi F)+\\frac{\\phi F}{3}R_{(g)},\\label{trace-gravity-eq-re}\n\\end{equation}\n\\begin{equation}\n\\Box\\phi-\\frac{\\partial_{\\mu}\\phi\\partial^{\\mu}\\phi}{4\\phi}=\\frac{1}{4\\omega}[8\\pi T-\\phi F R_{(g)}-\\frac{3}{2\\phi F}\\nabla_{\\mu}(\\phi F)\\nabla^{\\mu}(\\phi F)+3\\Box (\\phi F)].\\label{re-BD-equation}\n\\end{equation}\n We can see that Eqs. (\\ref{trace-gravity-eq-re}) and (\\ref{re-BD-equation}) describe the dynamics of the two scalar fields: $\\phi$ and $F$ in the Palatini-GBD theory, which are different from the results in the Palatini-$f(R)$ theory, where the scalar field carries no dynamics of its own \\cite{fr-review2}.\n\nIn this section, we derived the field equations of the GBD theory by using the non-standard Palatini approach, where the connection was treated as an independent dynamical variable. The gravitational field equations in this theory were gained by performing variations of action with respect to the metric and the connection, respectively. Variation with respect to the metric gave new field equation containing $F(\\tilde{R})$, and variation with respect to the connection gave the Riemann connection associated with metric $h_{\\mu\\nu}$ via appropriate conformal transformation. Based on the above field equations, we derive the linearized equations in the Palatini-GBD theory in the following.\n\n\n\n\n\\section{$\\text{ Linearized field equations in the Palatini-GBD theory}$}\n\n Modified gravitational theory should have the correct weak-field limit at the Newtonian and the post-Newtonian levels. In this section, we derive the linearized field equations in the Palatini-GBD theory via the weak-field approximation method. And then in the following two sections, we solve the linearized field equations for two cases: the vacuum case and the static point-mass case, respectively.\n\n\n As a begining, we discuss the weak-field approximations of GBD theory in the Palatini formalism via\n\\begin{eqnarray}\ng_{\\mu\\nu}=\\eta_{\\mu\\nu}+b_{\\mu\\nu},~~~~~~\\phi=\\phi_{0}+\\varphi,~~~~~~F=F_{0}+\\delta F,\\label{weak-conditions}\n\\end{eqnarray}\nwhere $\\eta_{\\mu\\nu}$ denotes the Minkowski metric, $\\phi$ and $F$ are two scalar fields, and the following three relations are required: $|b_{\\mu\\nu}|\\ll 1$, $|\\varphi|\\ll \\phi_{0}$ and $|\\delta F|\\ll F_{0}$. Using Eqs.(\\ref{Einstain-Tensor-Pala}-\\ref{weak-conditions}), linearized field equations in the Palatini-GBD theory are derived as\n \\begin{eqnarray}\n\\bar{R}_{(g)\\mu\\nu}-\\frac{\\bar{R}_{(g)}}{2}\\eta_{\\mu\\nu}=\n\\partial_{\\mu}\\partial_{\\nu}\\frac{\\delta F}{F_{0}}+\\partial_{\\mu}\\partial_{\\nu}\\frac{\\varphi}{\\phi_{0}}\n-\\eta_{\\mu\\nu}\\bar{\\Box}_{p}\\frac{\\delta F}{F_{0}}-\\eta_{\\mu\\nu}\\bar{\\Box}_{p}\\frac{\\varphi}{\\phi_{0}}+\\frac{8\\pi T_{\\mu\\nu}}{\\phi_{0}F_{0}},\\label{eq-weak-gravity}\n\\end{eqnarray}\n \\begin{eqnarray}\n\\bar{\\Box}_{p}\\varphi=\\frac{3}{4\\omega-3F_{0}}[8\\pi T-\\phi F_{0}\\bar{R}_{(g)}+3\\phi_{0}\\bar{\\Box}_{p}\\delta F],\\label{eq-weak-phi}\n\\end{eqnarray}\n \\begin{eqnarray}\n\\bar{\\Box}_{p}\\frac{\\delta F}{F_{0}}=\\frac{8\\pi T}{3\\phi_{0}F_{0}}-\\frac{\\bar{\\Box}_{p}\\varphi}{\\varphi_{0}}+\\frac{\\bar{R}_{(g)}}{3},\\label{eq-weak-Phi}\n\\end{eqnarray}\nwhere $\\bar{\\Box}_{p}=\\partial^{\\mu}\\partial_{\\mu}$. $\\bar{R}_{(g)\\mu\\nu}$ and $\\bar{R}_{(g)}$ denote the linearized quantities, and they can be rewritten as\n\\begin{eqnarray}\n\\bar{R}_{(g)\\mu\\nu}=\\frac{1}{2}(-2\\partial_{\\mu}\\partial\\nu b_{f}+2\\partial\\mu\\partial\\nu\\frac{\\varphi}{\\phi_{0}}-\\bar{\\Box}_{p}\\theta_{\\mu\\nu}+\\frac{\\eta_{\\mu\\nu}}{2}\\bar{\\Box}_{p}\\theta-\\eta_{\\mu\\nu}\\bar{\\Box}_{p} b_{f}+\\eta_{\\mu\\nu}\\bar{\\Box}_{p}\\frac{\\varphi}{\\phi_{0}}),\\label{linear-Rmunu}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\bar{R}_{(g)}=-3\\bar{\\Box}_{p} b_{f}+3\\bar{\\Box}_{p}\\frac{\\varphi}{\\phi_{0}}+\\frac{\\bar{\\Box}_{p}\\theta}{2}\\label{linear-R}\n\\end{eqnarray}\nvia introducing a new tensor $\\theta_{\\mu\\nu}=b_{\\mu\\nu}-\\frac{1}{2}\\eta_{\\mu\\nu}b-\\eta_{\\mu\\nu}\\frac{\\varphi}{\\phi_{0}}+\\eta_{\\mu\\nu}b_{f}$, where $b_{f}\\equiv \\frac{\\delta F}{F_{0}}$, $b=\\eta^{\\mu\\nu}b_{\\mu\\nu}$ and $\\theta=\\eta^{\\mu\\nu}\\theta_{\\mu\\nu}$.\nAfter choosing a so-called Lorenz gauge or the Harmonic gauge: $\\partial^{\\nu}\\theta_{\\mu\\nu}=0$ and using Eqs. (\\ref{eq-weak-gravity}-\\ref{linear-R}), we get the linearized gravitational field equation and the linearized scalar-field equations in the Palatini-GBD theory as follows\n\\begin{eqnarray}\n\\bar{\\Box}_{p}\\theta_{\\mu\\nu}=-\\frac{16\\pi T_{\\mu\\nu}}{\\phi_{0}F_{0}},\\label{eq-box-theta}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\bar{\\Box}_{p}\\varphi=\\frac{4\\pi T}{\\omega},\\label{eq-box-varphi}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\bar{\\Box}_{p} b_{f}=0,\\label{eq-box-Phi}\n\\end{eqnarray}\nwith $T=\\eta^{\\mu\\nu}T_{\\mu\\nu}$. Comparing the linearized field equation (\\ref{eq-box-Phi}) in the Palatini-GBD theory with that in the metric-GBD theory: $\\bar{\\Box}_{p} b_{f}-m_{s}^{2}b_{f}=\\frac{16\\pi\\omega T}{3\\phi_{0}F_{0}(2\\omega+3F_{0})}$, we can read some different properties for the scalar field $b_{f}$. In the Palatini-formalism of GBD theory, the scalar field $b_{f}$ is massless, while in the metric-formalism GBD the scalar field is massive. We also can see that the scalar field $b_{f}$ is source-free in the Palatini-formalism GBD, which is different from the result in the metric-formalism GBD.\n\n\n\\section{$\\text{Gravitational waves polarization in the Palatini-GBD theory}$}\n\n Gravitational-waves (GWs) physics is an important aspect for probing the viable gravitational theory. Studying on the polarization modes of GWs is also useful for exploring the valuable information on the early universe \\cite{GW-polar}. How many additional polarization modes are detected in GWs experiments could instruct us to study which theories of gravity. Given that more accurate observational data on GWs will be received in the future \\cite{GW-observations-future}, it is worthwhile to investigate GWs physics in alternative theories of gravity, especially in the Palatini-formalism of modified gravity. The weak-field approximation method provides a natural way to study the GWs. And in some references, the authors have applied this method to discuss the polarization of GWs in different theories \\cite{GW-other1,GW-other2,GW-other3,GW-other4,GW-other5,GW-other6,GW-other7,GW-other8}.\n\n Considering GWs which propagate along the $z$-direction, we have $k^{\\alpha}=\\varpi(1,0,0,1)$ with the angular frequency $\\varpi$. And let us consider an observer detecting the gravitational radiation described by a unit timelike vector: $u^{\\alpha}=(1,0,0,0)$. In the vacuum, we solve the wave Eqs. (\\ref{eq-box-theta}-\\ref{eq-box-Phi}) in the Palatini-formalism GBD to get\n\\begin{equation}\n\\theta_{\\mu\\nu}=A_{\\mu\\nu}(\\vec{p})\\exp(i k_{\\alpha}x^{\\alpha}),\\label{solution-theta}\n\\end{equation}\n\\begin{equation}\n\\varphi=c(\\vec{p})\\exp(i p_{\\alpha}x^{\\alpha}),\\label{solution-varphi}\n\\end{equation}\n\\begin{equation}\nb_{f}=d(\\vec{p})\\exp(i q_{\\alpha}x^{\\alpha}).\\label{solution-hf}\n\\end{equation}\nWhere $k_{\\alpha}$ denotes the four-wavevector, and it is a null vector with $\\eta_{\\mu\\nu}k^{\\mu}k^{\\nu}=0$. Eq.(\\ref{solution-theta}) denotes the plane-wave solution of gravitational radiation, while Eqs. (\\ref{solution-varphi}) and (\\ref{solution-hf}) denote the plane-wave solutions for the massless BD-field perturbation $\\varphi$ and the massless geometry-field perturbation $b_{f}$, respectively.\n\n\n\n\n\n\n In theory, several methods have been developed to analyze the polarization of GWs \\cite{gw-polar4,gw-polar5,gw-polar6,gw-polar1,gw-polar2,gw-polar3,GW-polar1a,GW-polar2a,GW-polar3a,GW-polar4a}, such as the Newman-Penrose (NP) method \\cite{gw-polar4,gw-polar5}, the geodesic deviation (GD) method \\cite{gw-polar1,gw-polar6}, etc. In the following, we investigate the polarization modes of GWs in the Palatini-GBD theory by using these two methods. In a local proper reference frame, the equation of geodesic deviation can be described as\n \\begin{equation}\n \\ddot{x}^{i}=-R^{i}_{~0k0}x^{k},\\label{eq-GD}\n \\end{equation}\n here $i$ and $k$ can be taken as $\\{1,2,3\\}$, respectively. $R^{i}_{~0k0}$ denotes the so-called \"electric\" components of the Riemann tensor with its expression as follows \\cite{GW-polar3a}\n \\begin{equation}\n R^{(1)}_{i0j0}=(h_{i0,0j}+h_{0j,i0}-h_{ij,00}-h_{00,ij}),\\label{linear-Riemann}\n \\end{equation}\n where $h_{\\mu\\nu}$ denotes the linear perturbation. Using Eqs. (\\ref{eq-GD}) and (\\ref{linear-Riemann}), we gain\n \\begin{eqnarray}\n \\ddot{x}(t)=-(xh_{11,00}+yh_{12,00}),~~~~~~~~\\nonumber\\\\\n \\ddot{y}(t)=-(xh_{12,00}+yh_{11,00}),~~~~~~~~\\nonumber\\\\\n \\ddot{z}(t)=(2h_{03,03}-h_{33,00}-h_{00,33})z.\\label{xyzdott}\n \\end{eqnarray}\n Using solution (\\ref{solution-theta}) and Eqs. (\\ref{xyzdott}), we obtain\n \\begin{eqnarray}\n \\ddot{x}(t)=k_{0}^{2}[\\hat{\\epsilon}^{(+)}(k_{0})x+\\hat{\\epsilon}^{(\\times)}(k_{0})y]\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,~~\\nonumber\\\\\n \\ddot{y}(t)=k_{0}^{2}[-\\hat{\\epsilon}^{(+)}(k_{0})y+\\hat{\\epsilon}^{(\\times)}(k_{0})x]\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,\\nonumber\\\\\n \\ddot{z}(t)=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\label{xyzdott-tensor}\n \\end{eqnarray}\nwhich describe the two standard plus and cross polarization modes of GR with the frequency $k_{0}$. For case of massless scalar field $\\varphi$ we have\n \\begin{eqnarray}\n \\ddot{x}(t)=-p_{0}^{2}xc(\\vec{p})\\exp(ip_{\\alpha}x^{\\alpha})+c.c.,~~~~\n \\ddot{y}(t)=-p_{0}^{2}yc(\\vec{p})\\exp(ip_{\\alpha}x^{\\alpha})+c.c.,~~~~\n \\ddot{z}(t)=0,\\label{xyzdott-bf}\n \\end{eqnarray}\nand for case of massless scalar field $b_{f}$ we obtain\n \\begin{eqnarray}\n \\ddot{x}(t)=-q_{0}^{2}xd(\\vec{p})\\exp(iq_{\\alpha}x^{\\alpha})+c.c.,~~~~\n \\ddot{y}(t)=-q_{0}^{2}yd(\\vec{p})\\exp(iq_{\\alpha}x^{\\alpha})+c.c.,~~~~\n \\ddot{z}(t)=0.\\label{xyzdott-varphi}\n \\end{eqnarray}\nObviously, Eqs.(\\ref{xyzdott-bf}) and (\\ref{xyzdott-varphi}) indicate a breathing type of GWs polarization, which have two oscillation modes with the frequency $q_{0}$ and frequency $p_{0}$, respectively. The same results can also be obtained by using the Newman-Penrose (NP) method \\cite{GW-polar3a,NP-tetrad}. Following the method shown in Refs. \\cite{GW-polar3a,GW-polar4a,GNP-polar-six1,GNP-polar-six2,GNP-polar-six3,GNP-polar-six4,GNP-polar-six5,GNP-polar-six6,NP-polar-six}, the amplitudes of six polarizations in the Palatini-GBD theory can be calculated as follows\n \\begin{eqnarray}\n p^{(l)}_{1}=-\\frac{1}{6}R_{0303}=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(x)}_{2}=-\\frac{1}{2}R_{0301}=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(y)}_{3}=\\frac{1}{2}R_{0302}=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(+)}_{4}=-R_{0101}+R_{0202}=-2k_{0}^{2}\\hat{\\epsilon}^{(+)}(k_{0})\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(\\times)}_{5}=2R_{0102}=2k_{0}^{2}\\hat{\\epsilon}^{(\\times)}(k_{0})\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(b)}_{6}=-R_{0101}-R_{0202}=2p_{0}^{2}c(\\vec{p})\\exp(ip_{\\alpha}x^{\\alpha})+2q_{0}^{2}d(\\vec{p})\\exp(iq_{\\alpha}x^{\\alpha})+c.c..\n \\label{p-np}\n \\end{eqnarray}\nHere the six polarizations modes of GWs are: the longitudinal scalar mode $p_{1}^{(l)}$, the vector-$x$ model $p_{2}^{(x)}$, the vecotr-$y$ mode $p_{3}^{(y)}$, the plus tensorial mode $p_{4}^{(+)}$, the cross tensorial mode $p_{5}^{(\\times)}$, and the breathing scalar mode $p_{6}^{(b)}$, respectively. Form expressions (\\ref{p-np}), we can read the plus tensor polarization mode, the cross tensor polarization mode, and a breathing scalar mode with two oscillation in the Palatini-GBD theory. It indicates that the extra scalar field $F$ and the BD scalar field generate the new polarizations of GWs which are not present in the standard GR or the Palatini-$f(\\tilde{R})$ theory. In the GR and the Palatini-$f(\\tilde{R})$ theory, both of them predict two tensorial polarization modes: $+$ and $\\times$, not have any scalar modes \\cite{GW-polar}.\n\n Comparing our results with other theoretical results in Refs. \\cite{GW-polar,GW-polar1a,GW-polar2a,GW-polar3a,GW-polar4a}, it is observed that the polarization modes of GWs in the Palatini-GBD theory are different from the results given in some other gravitational theories. For example, in the massive BD theory \\cite{GW-polar}, it has two standard tensorial modes of GR and two scalar modes (the longitudinal and the breathing modes); In the massless BD theory it owns two standard tensorial modes and one breathing scalar mode \\cite{GW-polar}; In the $f(R)$ theories, there are two tensorial modes and a massive scalar mode that is a mix of the longitudinal and the transverse breathing polarization \\cite{GW-polar1a,GW-polar2a}; In the $f(T,B)$ theory of teleparallel gravity (it is equivalent to $f(R)$ gravity by linearized the field equations in the weak field limit approximation), there are three polarizations \\cite{GW-polar3a}: the two standard of general relativity and an additional massive scalar mode, where the boundary term $B$ excites the extra scalar polarization; In the higher order local and non-local theories of gravity, they have three state of polarization and $n + 3$ oscillation modes \\cite{GW-polar4a} (concretely, they are the two transverse tensor ($+$) and ($\\times$) standard polarization modes of frequency $\\omega_{1}$, and the $n+1$ further scalar modes of frequency $\\omega_{2}, ..., \\omega_{n+2}$, each of which has the same mixed polarization, partly longitudinal and partly transverse).\n\n\n\n We also compare the results of GWs polarization in the Palatini-GBD theory with that in the metric-GBD theory. The plane-wave solutions in the metric-GBD theory can be expressed as \\cite{GBD-L}: $\\theta_{\\mu\\nu}=A_{\\mu\\nu}(\\vec{p})\\exp(i k_{\\alpha}x^{\\alpha})$, $\\varphi=a(\\vec{p})\\exp(i p_{\\alpha}x^{\\alpha})$, and $b_{f}=b(\\vec{p})\\exp(i q_{\\alpha}x^{\\alpha})$. Here, $\\varphi$ denotes the massless BD-field perturbation and $b_{f}$ denotes the massive geometry-field perturbation, respectively. For the massive plane wave propagating along $z-$direction, we have $q_{\\alpha}=(q_{0},0,0,q_{3})$ with $ m^{2}=q_{0}^{2}-q_{3}^{2}\\neq 0$. Originally, the NP formalism was applied to work out for massless waves. Recently, it was also generalized to explore the massive waves propagating along non-null geodesics \\cite{GW-polar3a,NP-tetrad}. Using this method, the non-zero amplitudes of polarizations for the metric-GBD theory in Ref. \\cite{GBD-L} are calculated as\n \\begin{eqnarray}\n p^{(l)}_{1}=\\frac{1}{6}(-q_{3}^{2}+q_{0}^{2})b(\\vec{p})exp(iq^{\\alpha}x_{\\alpha})+c.c.,~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(+)}_{4}=-\\sqrt{2}k_{0}^{2}\\hat{\\epsilon}^{(+)}(k_{0})\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(\\times)}_{5}=\\sqrt{2}k_{0}^{2}\\hat{\\epsilon}^{(\\times)}(k_{0})\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(b)}_{6}=2p_{0}^{2}a(\\vec{p})\\exp(ip_{\\alpha}x^{\\alpha})+2q_{0}^{2}b(\\vec{p})\\exp(iq_{\\alpha}x^{\\alpha})+c.c..\n \\label{p-np-metric}\n \\end{eqnarray}\n Obviously, Eqs.(\\ref{p-np-metric}) show that there are four polarizations modes for GWs in the metric-GBD theory: the two standard tensorial modes ($+$ and $\\times$), a scalar breathing mode with frequency $p_{0}$, and a massive scalar mode that is a mix of the longitudinal and the transverse breathing polarization with frequency $q_{0}$.\n\n\n\n\\section{$\\text{ PPN parameter in the Palatini-GBD theory}$}\n\n\nIn this section, we derive the theoretical expressions of the parametrized post-Newtonian (PPN) parameter $\\gamma$ in the Palatini-GBD theory by using the weak-field approximation method. Considering a static point-mass source, we have the form of the energy-momentum tensor: $T_{\\mu\\nu}=m_{p}\\delta(\\vec{r})diag(1,0,0,0)$. Obviously, the point particle is located at $\\vec{r}=0$. Solving Eqs. (\\ref{eq-box-theta}) and (\\ref{eq-box-varphi}), we get the perturbation solutions: $\\theta_{00}=\\frac{4m_{p}}{\\phi_{0}F_{0}}\\frac{1}{r}$ and $\\varphi(r)=\\frac{m_{p}}{\\omega}\\frac{1}{r}$. Combining relations: $b_{\\mu\\nu}=\\theta_{\\mu\\nu}-\\eta_{\\mu\\nu}\\frac{\\theta}{2}+\\eta_{\\mu\\nu}b_{f}-\\eta_{\\mu\\nu}\\frac{\\varphi}{\\phi_{0}}$ and $\\theta=\\eta^{\\mu\\nu}\\theta_{\\mu\\nu}=-\\frac{4m_{p}}{\\phi_{0}F_{0}}\\frac{1}{r}$, we gain the non-vanishing components of the metric perturbation\n\\begin{equation}\nb_{00}=\\frac{2m_{p}}{\\phi_{0}F_{0}r}+\\frac{m_{p}}{\\phi_{0}\\omega r}-b_{f},\\label{h00}\n\\end{equation}\n\\begin{equation}\nb_{ij}=\\frac{2m_{p}}{\\phi_{0}F_{0}r}-\\frac{m_{p}}{\\phi_{0}\\omega r}+b_{f}.\\label{hij}\n\\end{equation}\nHere $i,j =1,2,3$ is the space index. The term $\\frac{m_{p}}{\\phi_{0}\\omega r}$ in above two equations reflect the effect of scalar field $\\phi$. Considering that $b_{f}$ is negligible for a point-mass case, then the concrete form of the PPN parameter $\\gamma$ in the Palatini-GBD theory are derived as follows\n\\begin{equation}\n\\gamma=\\frac{b_{ii}}{b_{00}}=\\frac{2\\omega-F_{0}}{2\\omega+F_{0}}.\\label{gamma}\n\\end{equation}\n From Eq. (\\ref{gamma}), we can see the dependence of the PPN parameter $\\gamma$ with respect to model parameters: $\\omega$ and $F_{0}$. It is well known that a gravity theory alternative to GR should be tested by the well-founded experimental results. Some observations can be directly applied to constrain the value of the PPN parameter $\\gamma$. In Ref. \\cite{bound-omega-gamma}, the experimental bound on $\\gamma$ is: $|\\gamma-1|<2.3*10^{-5}$. For the Palatini-GBD theory, then we have that $\\gamma\\sim 1$ requires $\\omega \\gg F_{0}$, which can be consistent with the observational constraint: $ |\\omega| > 40000$ \\cite{bound-omega-gamma}.\n\n\n\n\\section{$\\text{Conclusions}$}\n\nSeveral observational and theoretical problems motivate us to investigate the modified or alternative theories of GR. Lots of modified gravity theories have been proposed and widely studied. In the BD modified theory, the scalar field can be introduced by considering a time-variable Newton gravity constant. Many extended versions of the BD theory have been explored and developed. As one of the generalized BD theories, some interesting properties have been found in the metric-formalism of GBD theory \\cite{GBD,GBD-L}. For examples: (1) In the GBD model, the state parameter of geometrical dark energy can cross over the phantom boundary $w=-1$ as achieved in the quintom model, without bearing the problems existing in the two-fields quintom model, such as the problem of negative kinetic term and the fine-tuning problem, etc. (2) It is well known that the metric-$f(R)$ theories are equivalent to the BD theory with a potential (abbreviated as BDV) for taking a specific value of the BD parameter $\\omega=0$, where the specific choice: $\\omega=0$ for the BD parameter is quite exceptional, and it is hard to understand the corresponding absence of the kinetic term for the field. However, for the GBD theory, it is similar to the double scalar-fields model, and it owns the non-disappeared kinetic term in the action. In addition, the GBD theory tends to investigate the physics from the viewpoint of geometry, while the BDV or the two scalar-fields quintom model tends to solve physical problems from the viewpoint of matter. It is possible that several special characteristics of scalar field could be revealed through studies of geometrical gravity in the GBD.\n\n\nIn this paper, we studied the generalized Brans-Dicke theory in the Palatini formalism. Firstly, we derived the field equations for the gravitational field, the independent connection and the BD scalar field, respectively. Secondly, using the weak-field method we obtained the linearized gravitational field equation and the linearized scalar-fields equations. We showed various properties of the geometrical scalar field in the Palatini-formalism of GBD theory: it is massless and source-free, which are different from the results given in the metric-formalism of GBD theory. According to the weak-field equations, we investigated the parameterized post Newton parameter in the Palatini-GBD theory by using the point-source method. It was shown that $\\gamma=\\frac{2\\omega-F_{0}}{2\\omega+F_{0}}\\sim 1$ requires $\\omega \\gg F_{0}$, which can be consistent with the observational constraint: $ |\\omega| > 40000$. Comparing to the Ref. \\cite{GBD-L}, we can see that the difference for expressions of $\\gamma$ between the Palatini-GBD theory and the metric-GBD theory. In the metric-GBD theory \\cite{GBD-L}, $\\gamma$ depends on the parameters: $\\omega$, $F_{0}$ and $m_{s}$. Thirdly, we discussed the gravitational waves physics in the Palatini-GBD theory. The properties of GWs in the modified gravity theory have recently attracted lots of attention \\cite{gw-mg1,gw-mg2,gw-mg3,gw-mg4,gw-mg5}, since investigating the polarization modes of GWs can serve to discriminate the different gravitational theories.\nThe results in the Palatini-formalism of GBD showed that the extra scalar field $F$ and the BD scalar field generated the new polarizations for GWs which were not present in the standard GR or the Palatini-$f(R)$ theory, where both theories (GR and Palatini-$f(\\tilde{R})$) predicted two tensorial polarization modes: $+$ and $\\times$, not have any scalar modes. Concretely, we can read that there are three modes of polarization and four oscillation for GWs in the Palatini-GBD theory, i.e. the plus tensor polarization mode, the cross tensor polarization mode, and a breathing scalar mode with two oscillation. The results of GWs polarization in the Palatini-GBD theory are different from that in the metric-GBD theory. In the metric-GBD theory, there are four polarizations modes: the two standard tensorial modes ($+$ and $\\times$), a scalar breathing mode, and a massive scalar mode that is a mix of the longitudinal and the breathing polarization.\n\n\n\n\n\\textbf{\\ Acknowledgments }\n The research work is supported by the National Natural Science Foundation of China (11645003,11705079), and supported by LiaoNing Revitalization Talents Program.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n An intuitive understanding of a prototype of non-Fermi liquid (NFL), a multichannel Kondo model, was proposed by Nozi\\`eres and Blandin about 25 years ago.\\cite{NB} The understanding of the multichannel Kondo model was obtained by various methods, such as Bethe-ansatz\\cite{Wiegman, Andrei}, boundary conformal field theory (BCFT)\\cite{AffLud1,AffLud2,AffLud3,AffLud4,2IK}, numerical renormalization group (NRG)\\cite{Cragg,Pang}, and so on. In particular, we can compare the NRG finite size spectra and those of the BCFT even if we have NFL spectra and the agreements are excellent.\\cite{Kim2,AffLud4,2IK} \n\n In the realistic model of diluted f-electron systems, a candidate for a two-channel Kondo (2CK) model is the quadrupolar Kondo model\\cite{Cox} for U and Ce based alloys, in which non-Kramers doublet in $f^2$ configuration plays an important role. Several models that have a NFL property were proposed, in which the $\\Gamma_8$ conduction electrons interact with a localized $f^1$ or $f^2$ crystalline-electric-field state under cubic symmetry\\cite{Koga1,KusuKura,Koga2,Kim1}. These models may be relevant for dilute alloys of Ce$^{3+}$ or U$^{4+}$ ions, such as La$_{1-x}$Ce$_x$Cu$_2$Si$_2$\\cite{Andraka}, UCu$_{5-x}$Pd$_x$\\cite{Andrade} among others. In general, NFL behaviors are observed in real materials as transient phenomena, which are controlled by temperature, pressure, alloying and so on.\n\n In this Letter, we reexamine an impurity Kondo model in which a localized f-electron with $\\Gamma_8$ symmetry interacts with $\\Gamma_8$ conduction electrons under cubic symmetry. This model was proposed as that of Ce$_x$La$_{1-x}$B$_6$\\cite{Koga1}. The earlier NRG calculations confirmed the existence of a NFL fixed point that is unstable against the particle-hole (PH) symmetry breaking. Indeed, Ce$_x$La$_{1-x}$B$_6$ is not located near this NFL fixed point, so that the property of this NFL may not be relevant to the case of Ce$_x$La$_{1-x}$B$_6$. However, the NFL would be realized in the system with small quadrupolar interactions that break PH symmetry. Theoretically, the origin of the NFL was thought to be mysterious from the conventional BCFT point of view. The NRG energy spectrum was similar to that of 2CK model but the origin was unknown.\n\nAlthough it is difficult to control PH asymmetry in experiments, for both the pursuit of new materials and theoretical interest, it is worthwhile to study the detailed properties of this NFL by BCFT. It is noted that a BCFT approach is a powerful tool for answering {\\it why} NFL behaviors emerge in various impurity problems. The same situation occurs in the case of the two-impurity Kondo model (the NFL is unstable against PH asymmetry of conduction electrons), which has been extensively investigated so far\\cite{Jones}. Roughly speaking, the quantum critical phenomena near the antiferromagnetic critical point at zero temperature\\cite{Stewart} can be seen as a kind of a lattice generalization of the two-impurity Kondo model (however, it is not so simple). The PH asymmetry in a generalized 2CK model\nwas also discussed\\cite{Pan2}. In this case, in the presence of a double tensor interaction that breaks the PH symmetry, the system flows to a Fermi liquid. \n\n\n We assume the following points for the local f-electron state: (i) cubic symmetry, (ii) $f^1$ $\\Gamma_8$ ground state configuration, (iii) the lowest excited state is f$^0$ and\/or $f^2$ $\\Gamma_1$ singlet configuration. \n Under these assumptions, we can map the $\\Gamma_8$ index to a pseudospin-$3\/2$ representation as $|\\Gamma_{8,{\\pm \\frac{3}{2}}}\\rangle=\\mp(| \\pm \\frac{3}{2}\\rangle+\\sqrt{5}|\\mp \\frac{5}{2}\\rangle)\/\\sqrt{6}$ and $|\\Gamma_{8,{\\pm \\frac{1}{2}}}\\rangle=\\pm| \\pm \\frac{1}{2}\\rangle$, where $|j_z\\rangle$ is a state with the total angular momentum $J=5\/2$ and its z-component $j_z$.\n\n\n\nWe start with the following pseudospin-$3\/2$ Kondo model (see the derivation in ref. 13) in 1-dimension considering the s-wave scattering at the impurity site\\cite{AffLud2}:\n\\begin{eqnarray}\nH&=&H_0+\\sum_{m=\\rm dip,quad,oct}H_{m},\\label{H}\\\\\nH_0&=&\\frac{iv_F}{2\\pi}\\sum_{\\mu=\\pm \\frac{3}{2},\\pm\\frac{1}{2}}\\int dx\\psi_{\\mu}^{\\dagger}(x)\\partial \\psi_{\\mu}(x),\\\\\nH_{m}&=& J_{m}\\sum_{\\mu,\\nu=\\pm \\frac{3}{2},\\pm\\frac{1}{2}}\\psi_{\\mu}^{\\dagger}(0)({\\bf x}_c^m)_{\\mu\\nu}\\psi_{\\nu}(0)\\cdot {\\bf X}_I^m,\n\\end{eqnarray}\nwhere we introduce left-moving fermion annihilation operators $\\psi_{\\mu}(x)$, Fermi velocity $v_F$ and spin $3\/2$ dipolar (${\\bf x}_c^{\\rm dip}={\\bf s}_c$), quadrupolar (${\\bf x}_c^{\\rm quad}={\\bf q}_c$) and octupolar (${\\bf x}_c^{\\rm oct}={\\bf t}_c$) matrices of the conduction electron (similar definitions for ${\\bf X}_I^m$ of the impurity).\\cite{Relation} $J_{m}$ is the coupling constant of each multipoles. It is noted that under the assumptions (i)-(iii) the interactions are isotropic in the pseudospin space.\n\n\nThe ``conformal embedding'' often found in literatures is SU(4)$\\to$ SU(2)$_{10}\\otimes$ SU(2)$_4$.\\cite{Kim2}\n The SU(2)$_{10}$ corresponds to a spin current $ {{ \\mbox{\\boldmath $ \\mathcal J$}}}(x)$\\cite{J}, i.e., {\\it dipole}. The other SU(2) current, SU(2)$_4$, is an axial charge (AC) current $\\mbox{\\boldmath $\\mathcal Q$}(x)$\\cite{Q}. In this embedding, $H_0$ can be written in the following Sugawara form:\n\\begin{eqnarray}\n\\frac{l}{\\pi v_F}H_0=\\sum_{n=-\\infty}^{\\infty}\\Big(\\frac{1}{4}:\\mbox{\\boldmath $\\mathcal Q$}_{-n}\\cdot\\mbox{\\boldmath $\\mathcal Q$}_{n}: + \\frac{1}{12}:{\\mbox{\\boldmath $\\mathcal J$}}_{-n}\\cdot \n{\\mbox{\\boldmath $\\mathcal J$}}_{n}:\\Big), \\label{Hboson}\n\\end{eqnarray}\nwhere \n${\\mbox{\\boldmath $\\mathcal J$}}_{n}\\ {\\rm and}\\ {\\mbox{\\boldmath $\\mathcal Q$}}_{n}$ are the Fourier components of ${\\mbox{\\boldmath $\\mathcal J$}}(x)\\ {\\rm and}\\ {\\mbox{\\boldmath $\\mathcal Q$}}(x)$, respectively. We set the system size to $2l$. $:A:$ indicates the normal ordering of the operator $A$. The energy eigenvalues of the right hand side of eq. (\\ref{Hboson}) for primary states that form conformal towers\\cite{Itz} are given by\n\\begin{eqnarray}\nE(q,j)=\\frac{q(q+1)}{4}+\\frac{j(j+1)}{12},\\label{E0}\n\\end{eqnarray}\n where $(q=0,1\/2, 1)$ and $(j=0,1\/2, \\cdots, 5)$. The energies of the descendant states\\cite{Itz} are given by $E(q,j)+n$, where $n$ is a positive integer and indicates the PH excitations from the ground state.\\cite{AffLud3} The descendant states generally have different quantum numbers from those of the corresponding primary state in the same conformal tower.\n The form of eq. (\\ref{Hboson}) is suitable to the case of $J_{\\rm quad}=J_{\\rm oct}=0$. In this case, the impurity spin can be absorbed into ${\\mbox{\\boldmath $\\mathcal J$}}_{n}$\\cite{AffLud1} and the predicted NFL spectra are in complete agreement with the NRG results\\cite{Kim2,Koga1,Koga2}.\n\nIn the presence of quadru- and octupolar interactions, the above SU(2)$_{10}$ absorption is not applicable.\nAs noted by Wu {\\it et al.} the spin $3\/2$ fermionic system has an exact SO(5) symmetry under some conditions\\cite{Wu1}. In the following, we transform the Hamiltonian (\\ref{H}) into the SO(5) language, i.e., the embedding is SU(4)$\\to $SO(5)$_2\\otimes$ SU(2)$_4$.\n\nFirst, quadrupolar matrices of $s_c=3\/2$ have the following forms:\n\\begin{eqnarray}\nq_c^{3z^2-r^2}&\\equiv&\\frac{1}{\\sqrt{3}}(3s_c^zs_c^z-\\frac{3}{2}\\frac{5}{2}\\hat{\\bf 1})\\equiv \\sqrt{3}\\Gamma^4 \\label{qz},\\\\\nq_c^{x^2-y^2}&\\equiv&\\frac{1}{\\sqrt{2}}(s_c^xs_c^x-s_c^ys_c^y)\\equiv-\\sqrt{3}\\Gamma^5,\\\\\nq_c^{xy}&\\equiv&s_c^xs_c^y+s_c^ys_c^x\\equiv -\\sqrt{3}\\Gamma^1,\\\\\nq_c^{yz}&\\equiv&s_c^ys_c^z+s_c^zs_c^y\\equiv\\sqrt{3}\\Gamma^3,\\\\\nq_c^{zx}&\\equiv&s_c^zs_c^x+s_c^xs_c^z\\equiv\\sqrt{3}\\Gamma^2,\\label{qzx}\n\\end{eqnarray}\nwhere $\\hat{\\bf 1}$ is an identity matrix and the spin $3\/2$ operators $s_c^i\\ (i=x,y,z)$ are constructed on the basis of $^{\\dagger}(\\psi_{\\frac{3}{2}},\\psi_{\\frac{1}{2}},\\psi_{\\frac{-1}{2}},\\psi_{\\frac{-3}{2}})$. We have introduced five Dirac matrices $\\Gamma^a\\ (1\\le a \\le 5)$.\nFrom eqs. (\\ref{qz})-(\\ref{qzx}), we can define ten SO(5) generators $\\Gamma^{ab}$ as $\\Gamma^{ab}\\equiv \\frac{1}{2i}[\\Gamma^a, \\Gamma^b]$. These generators satisfy the SO(5) commutation relations:\n\\begin{eqnarray}\n[\\Gamma^{ab}, \\Gamma^{cd}]=-2i(\\delta_{bc}\\Gamma^{ad}-\\delta_{ac}\\Gamma^{bd}-\\delta_{bd}\\Gamma^{ac}+\\delta_{ad}\\Gamma^{bc}).\n\\end{eqnarray}\n\nThe important point is that $H_{\\rm dip}$ and $H_{\\rm oct}$ are written by the ten $\\Gamma^{ab}$ when the condition $J_{\\rm dip}=J_{\\rm oct}$ is satisfied. Thus, under this condition, each multipolar Hamiltonian can be written in the SO(5)'s vector $n^a_I$ ($n_c^a$) and generators $L^{ab}_I$ ($L_c^{ab}$) of the impurity (conduction electron), as follows:\n\\begin{eqnarray}\nH_{\\rm quad}&=& 12J_{\\rm quad}\\sum_{a=1}^5n_c^a(0)n^a_I,\\label{Hq}\\\\\nH_{\\rm dip}+H_{\\rm oct}&=& 5J_{\\rm dip}\\sum_{a=1}^5\\sum_{b>a}^5L_c^{ab}(0)L^{ab}_I,\\label{Hdo}\n\\end{eqnarray}\n where \n\\begin{eqnarray}\nn_c^a(x)&\\equiv& \\frac{1}{2}\\sum_{\\mu\\,\\nu=\\pm \\frac{3}{2},\\pm\\frac{1}{2}}\\psi_{\\mu}^{\\dagger}(x)(\\Gamma^a)_{\\mu\\nu}\\psi_{\\nu}(x),\\label{nc}\\\\\nL_c^{ab}(x)&\\equiv&-\\frac{1}{2}\\sum_{\\mu,\\nu=\\pm \\frac{3}{2},\\pm\\frac{1}{2}}\\psi^{\\dagger}_{\\mu}(x)(\\Gamma^{ab})_{\\mu\\nu}\\psi_{\\nu}(x).\\label{Lc}\n\\end{eqnarray}\n\\begin{table}[t]\n\t\\begin{tabular}{cccc}\n \\hline\n $q$ & $j$ & {\\bf u} & $E(q,{\\bf u}) \\ (E(q,j))$\\\\\n \\hline\n \\hline\n $0$ & $0$ & $\\bf 1$ & $0$\\\\\n \\hline\n $1\/2$ & $3\/2$ & $\\bf 4$ & $1\/2$\\\\\n \\hline\n $1$ & $2$ & $\\bf 5$ & $1$\\\\\n \\hline\n $0$ & $3$ & $\\bf 10$ & $1$\\\\\n $0$ & $1$ & & $1$\\\\\n \\hline\n $1$ & $0$ & $\\bf 1$ & $1$\\\\\n \\hline\n $3\/2$ & $3\/2$ & $\\bf 4$ & $3\/2$\\\\\n $1\/2$ & $3\/2$ & $\\bf 4$ & $3\/2$\\\\\n \\hline\n $1\/2$ & $7\/2$ & $\\bf 16$ & $3\/2$\\\\\n $1\/2$ & $5\/2$ & & $3\/2$\\\\\n $1\/2$ & $1\/2$ & & $3\/2$\\\\\n \\hline\n \\end{tabular}\n\\caption{Low-energy spectrum of free Hamiltonian for nondegenerate ground state. The first and the third columns are the AC and SO(5) indices, respectively. The second column is the eigenvalue of the spin current in eq. (\\ref{E0}). $E(q,{\\bf u})$ and $E(q,j)$ are measured from the ground state using eqs. (\\ref{E1}) and (\\ref{E0}), respectively.}\n\\label{tbl-1}\n\\end{table}\n\n\nThe free Hamiltonian $H_0$ in (\\ref{H}) is expressed using the SO(5) generators\n\\begin{eqnarray}\n\\frac{l}{\\pi v_F}H_0=\\sum_{n=-\\infty}^{\\infty}\\Big(\\frac{1}{4}:\\mbox{\\boldmath $\\mathcal Q$}_{-n}\\cdot\\mbox{\\boldmath $\\mathcal Q$}_{n}: + \\frac{1}{8}:{\\mbox{\\boldmath $\\mathcal L$}}_{-n}\\cdot \n{\\mbox{\\boldmath $\\mathcal L$}}_{n}:\\Big), \\label{Hboson2}\n\\end{eqnarray}\nwhere $({\\mbox{\\boldmath $\\mathcal L$}}_{n})^{ab}$ is the n-th Fourier component of SO(5) current $L_c^{ab}(x)$: ${\\mbox{\\boldmath $\\mathcal L$}_n}=({\\mathcal L^{12}_{n}}, {\\mathcal L^{13}_{n}},\\cdots ,{\\mathcal L^{35}_{n}}, {\\mathcal L^{45}_{n}})$ and satisfies a level 2 SO(5) Kac-Moody algebra. The energy eigenvalues $E(q,{\\bf u})$ for the primary states are expressed by the Casimir operators in each sector\n\\begin{eqnarray}\nE(q,{\\bf u})=\\frac{q(q+1)}{4}+\\frac{\\bar{c}_{\\bf u}}{8},\\label{E1}\n\\end{eqnarray}\nwhere $q=0,1\/2, 1$ for the AC sector (this is as same as in eq. (\\ref{E0})), and $\\bf u=1\\ {\\rm(identity)},\\ 4\\ {\\rm (spinor)}\\ ,5\\ {\\rm (vector)}$ for the SO(5) sector. ${\\bf u}$ indicates the dimension of the representation in the SO(5), see Fig. \\ref{fig-1}. There are three primary fields in both the AC and the SO(5) sectors.\nThe values of $\\bar{c}_{\\bf u}$ are $\\bar{c}_{\\bf 1}=0,\\ \\bar{c}_{\\bf 4}=5\/2\\ {\\rm and }\\ \\bar{c}_{\\bf 5}=4$. The noninteracting energy spectra for the nondegenerate ground state are shown in Table \\ref{tbl-1} together with the indices of $j$ in eq. (\\ref{E0}). We can see that both eqs. (\\ref{E0}) and (\\ref{E1}) can reproduce the spectrum of free $s_c=3\/2$ fermions.\n\n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=.25\\textwidth]{.\/fig1.eps}\n \\end{center}\n\\caption{$H_1-H_2$ diagrams associated with the SO(5) multiplies that enter the low-energy spectrum, where $2H_1\\equiv \\Gamma^{15}$ and $2H_2\\equiv \\Gamma^{23}$ forming the Cartan subalgebra of SO(5) group. In (a), the corresponding components of the spinor are indicated.}\n\\label{fig-1}\n\\end{figure}\n\n\nNext, we consider eqs. (\\ref{Hq}), (\\ref{Hdo}) and (\\ref{Hboson2}) together. It is noted that the total Hamiltonian (\\ref{H}) is written as\n\n\\begin{eqnarray}\n\\frac{l}{\\pi v_F}H&=&\\sum_{n=-\\infty}^{\\infty}\\Big(\\frac{1}{4}:\\mbox{\\boldmath $\\mathcal Q$}_{-n}\\cdot\\mbox{\\boldmath $\\mathcal Q$}_{n}: + \\frac{1}{8}:{\\mbox{\\boldmath $\\mathcal L$}}_{-n}^{'}\\cdot \n{\\mbox{\\boldmath $\\mathcal L$}}_{n}^{'}:\\Big)\\nonumber\\\\\n&&+\\frac{12lJ_{\\rm quad}}{\\pi v_F}\\sum_{a=1}^5n_c^a(0)n^a_I+\\rm const.,\\label{HH}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\mathcal L'}_{n}^{ab}\\equiv{\\mathcal L}_{n}^{ab}+\\frac{20lJ_{{\\rm dip}}}{\\pi v_F}L_I^{ab}.\n\\end{eqnarray}\nThe values at possible fixed points, the $J_{\\rm dip}^*$, $J^*_{\\rm quad}$ and $J^*_{\\rm oct}$ are determined (assumed) as\n\\begin{eqnarray}\n J_{\\rm dip}^*=J_{\\rm oct}^*=\\frac{\\pi v_F}{20l}{\\rm,\\ \\ \\ \\ and}\\ \\ \\ J^*_{\\rm quad}=0.\\label{Jc}\n\\end{eqnarray}\n The case in which $J_{\\rm quad}\\not=0$ is discussed later. At these values of the couplings, the impurity SO(5) ``superspin'' can be absorbed into the conduction electron ``superspin'' current, generating the new SO(5) ``superspin'' $ {\\mbox{\\boldmath $\\mathcal L$}}_{n}^{'}$. This impurity absorption does not affect the SO(5) Kac-Moody algebra, except the gluing conditions in Table \\ref{tbl-1}, which is much the same as in the case of the multichannel Kondo model\\cite{AffLud1}. \n\n\n At this stage, there is a need to introduce a suitable fusion rule to generate the nontrivial spectrum. Because the interaction is only in the SO(5) sector, any fusion in the AC sector is unphysical. An important point is that the impurity is described in the spin $3\/2$ representation (i.e., dimension 4). We find that the desired fusion is a direct product of $\\bf 4$ representation in the SO(5) sector to each state in the spectra of the free Hamiltonian. To execute this, the following formulae are useful (easily deducible from Fig. \\ref{fig-1}): $\n\\bf 1\\otimes 4= \\bf 4,\\ \n\\bf 4\\otimes 4= \\bf 1\\oplus 5\\oplus 10\\ {\\rm and}\\ \n\\bf 4\\otimes 5= \\bf 4\\oplus 16$.\n\nThus, even after absorbing the impurity ``superspin'', we can calculate the energy spectra at this fixed point by using eq. (\\ref{E1}\n. \nThe obtained spectra are shown in Table \\ref{tbl-2}(a). As shown by previous NRG studies\\cite{Koga1, KusuKura}, the NFL spectra are the same as those of the 2CK NFL if the AC sector in the present model and the spin sector in the 2CK model are interchanged. Indeed, the existence of a SO(5) symmetry in the 2CK model has been pointed out before in literatures.\\cite{AffLud4}\n\n\\begin{table}[t]\n\\begin{center}\n\t\\begin{tabular}{cccccccc}\n \\hline\n (a)&$q$ & {\\bf u} & $E(q,{\\bf u})$ & (b) & $q$ & {\\bf u} & $\\Delta$\\\\\n \\hline\n \\hline\n &$1\/2$ & $\\bf 1$ & $0$ & & $0$ & $\\bf 1$ & $0$ \\\\\n &$0$ & $\\bf 4$ & $1\/8$ & & $0$ & $\\bf 5$ & $1\/2$ \\\\\n &$1\/2$ & $\\bf 5$ & $1\/2$ & & $1\/2$ & $\\bf 4$ & $1\/2$ \\\\\n &$1$ & $\\bf 4$ & $5\/8$ & & $1\/2$ & $\\bf 4$ & $1\/2$ \\\\\n &$3\/2$ & $\\bf 1$ & $1$ & & $1$ & $\\bf 1$ & $1\/2$ \\\\\n &$1\/2$ & $\\bf 10$ & $1$ & & $1$ & $\\bf 5$ & $1$ \\\\\n &$1\/2$ & $\\bf 1$ & $1$ & & $0$ & $\\bf 10$ & $1$ \\\\\n \\hline\n \\end{tabular}\n\\caption{(a) Low energy spectrum of NFL fixed point with $E(q,{\\bf u})\\le 1$. In the third column, the ground state energy $3\/16$ is subtracted. (b) Operator contents at the NFL fixed point. We show the operators with $\\Delta \\le 1$ where $\\Delta$ means the scaling dimension of the corresponding operator. \n}\n\\label{tbl-2}\n\\end{center}\n\\end{table}\n\n\nThe operator contents at the NFL fixed point are obtained by applying a double fusion\\cite{AffLud1} in the SO(5) sector (see Table \\ref{tbl-2}(b)). Again, we obtain the same scaling dimensions $\\Delta$ as those at the 2CK fixed point with the above-mentioned interchange. The operator with $(q, {\\bf u})=(0,{\\bf 5})$ is the SO(5) vector $\\mbox{\\boldmath $\\mathcal \\phi$}_{\\rm SO(5)}$, which corresponds to the quadrupolar operator in the original spin $3\/2$ representation. This operator has $\\Delta=1\/2$, so that the corresponding local quadrupolar susceptibility diverges logarithmically at low temperatures. This is consistent with the result of NRG studies \\cite{Koga1, KusuKura}. The local charge and pair field susceptibility would diverge at low temperatures, because the operator $(q, {\\bf u})=(1,{\\bf 1})$, $\\mbox{\\boldmath $\\mathcal \\phi$}_{\\rm AC}$, which is the AC vector, has $\\Delta=1\/2$. The dipolar susceptibility in addition to the octupolar one, is classified in $\\bf 10$ representation with $\\Delta=1$. This means that the dipolar and the octupolar susceptibilities $\\chi$ are not singular, which is also consistent with the NRG calculation\\cite{Koga1,KusuKura}.\n\n\nNext we discuss the stability of the NFL fixed point against various perturbations.\n\na) In the presence of a uniaxial distortion, the term $h_Qn_I^4$, which breaks the SO(5) symmetry, appears in the effective Hamiltonian. This term allows $\\mbox{\\boldmath $\\mathcal \\phi$}_{\\rm SO(5)}$ with $\\Delta=1\/2$ to appear. Thus, the conjugate field $h_Q$ becomes a relevant perturbation and the NFL fixed point becomes unstable.\n\nb) In the presence of a potential scattering at the impurity site $V\\sum_{\\mu}\\psi_{\\mu}^{\\dagger}(0)\\psi_{\\mu}(0)=VQ_z(0)+\\ \\rm const.$, the SU(2) symmetry in the AC sector is broken. In this case, $\\phi_{\\rm AC}^3$, the component $q_z=0$ of $\\mbox{\\boldmath $\\mathcal \\phi$}_{\\rm AC}$ with $\\Delta=1\/2$, can appear in the effective Hamiltonian. Again, the NFL fixed point becomes unstable. The PH symmetry can also be broken by the quadrupolar interaction of eq. (\\ref{Hq}). Indeed, the system flows into the Fermi liquid fixed point of SU(4) Coqblin-Schrieffer model without PH symmetry\\cite{Koga1}. \n\nc) The exchange anisotropy in the SO(5) sector is irrelevant, because the marginal operator $(q,{\\bf u})=(0,{\\bf 10})$ cannot appear under the time-reversal symmetry as discussed in Ref. 8 (that is, our assumption $J_{\\rm dip}=J_{{\\rm oct}}$, does not affect the present result).\\cite{COMMENT}\n\n\\begin{figure}[t!]\n \\begin{center}\n \\includegraphics[width=.4\\textwidth]{.\/fig2.eps}\n \\end{center}\n\\caption{$C_{\\rm imp}\/T$ vs $T$ and ${\\rm Im}\\chi_c(\\omega)$ vs $\\omega$. $V$ is the strength of potential scattering at the impurity site. The parameters used are $J_{\\rm dip}=J_{\\rm oct}=0.1D$ and $J_{\\rm quad}=0.0D$, where $D$ is the half of the bandwidth of conduction electrons.}\n\\label{fig-2}\n\\end{figure}\n\nIn early NRG studies\\cite{Koga1,KusuKura}, the NFL fixed point was considered to be similar to the 2CK fixed point. But the question arises, `` Is the present NFL really equivalent to the 2CK ?'' To verify this, we consider the effective Hamiltonian near the NFL fixed point. If the present NFL and the NFL of the 2CK were equivalent, the leading irrelevant operator would be $\\mbox{\\boldmath $\\mathcal Q$}_{-1}\\cdot\\mbox{\\boldmath $\\mathcal \\phi$}_{\\rm AC}$ with $\\Delta=1\/2+1=3\/2$. This operator, however, is not physically adequate because the impurity absorption occurs in the SO(5) sector. The low-energy effective Hamiltonian should be made by using operators in the SO(5) sector and these should transform as a singlet\\cite{AffLud1}. \n\nBecause ${\\mathcal L}^{ab}_{-1}\\phi^c_{\\rm SO(5)}\\ (1\\le a,b,c\\le 5)$ type operators with $\\Delta=3\/2$ cannot become a singlet, the leading irrelevant operator should be the energy-momentum operator $\\sum_{ab}:L_c^{ab}(0)L_c^{ab}(0):$ with $\\Delta=2$. This leads us to an important conclusion: {\\it impurity specific heat does not diverge at low temperatures} unlike in the 2CK case, {\\it but shows a linear temperature behavior}. As we mentioned above, $\\chi$ is not singular, so the Wilson ratio $R_W=(\\delta\\chi\/\\chi)\/(\\delta C\/C)$ is calculated using the conformal charge in each of sectors $c_{\\rm SO(5)}$ and $c_{\\rm AC}$ as $R_W=(c_{\\rm SO(5)}+c_{\\rm AC})\/c_{\\rm SO(5)}=(5\/2+3\/2)\/(5\/2)=8\/5$($C$: specific heat).\\cite{AffLud3} This result is different from that of the 2CK case: $R_W=8\/3$.\n\nFinally, to check the new results above, we show the results of the NRG calculations of the impurity specific heat $C_{\\rm imp}$ and the z-component of the dynamical AC susceptibility for localized conduction electrons $\\chi_c(\\omega)$ at zero temperature in Fig. {\\ref{fig-2}}. Note that $\\chi_c(\\omega)$ is the dynamical charge susceptibility of the conduction electrons at the impurity site. We used logarithmic discretization parameter $\\Lambda=3$ and kept 400 states at each NRG step. It was confirmed that $C_{\\rm imp}\/T$ is constant at low temperatures and that Im$\\chi(0)\\not=0$ indicates the divergence of the conduction electron's charge susceptibility. We can see that the divergence of the charge susceptibility is suppressed when PH symmetry is broken ($V\\not=0$ case). We also obtained the residual impurity entropy $S_0=k_B\\log \\sqrt{2}$, which is the same value as that at the 2CK NFL fixed point. An increase in the data $V\/D=10^{-4}$ of the $C_{\\rm imp}\/T$ is the natural consequence of the entropy release from $k_B\\log \\sqrt{2}$ to $0$.\n\nIn summary, we have investigated a multipolar Kondo model with $S_I=3\/2$ and $s_c=3\/2$ using BCFT and NRG. The 2CK-like NFL fixed point observed in the earlier NRG calculation is explicitly derived using the ``superspin'' absorption in a hidden symmetry SO(5). We find that the leading irrelevant operator at the NFL fixed point is ``Fermi liquid-like'' in contrast to its 2CK-like NFL spectra. All predictions are consistent with earlier works and the present NRG calculation. In particular, the low temperature impurity specific heat is proportional to temperature, the Wilson ratio $R_W=8\/5$ and the local charge susceptibility of conduction electrons diverges at zero temperature. These results remarkably distinguish the present NFL from the 2CK model.\n\n\\vspace{.5cm}\nThe author would like to thank A. Yotsuyanagi, T. Takimoto, H. Kohno and K. Miyake for valuable comments. This work was supported by the 21st Century COE Program (G18) of the Japan Society for the Promotion of Science.\nThe author is supported by the Research Fellowships of JSPS for Young Scientists.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nIn the face of the proliferation of real-time IoT applications, fog computing has come as a promising complement to cloud computing by extending cloud to the edge of the network to meet the stringent latency requirements and intensive computation demands of such applications \\cite{xiao2017qoe}.\n\nA typical fog computing system consists of a set of geographically distributed fog nodes\nwhich are deployed at the network periphery with elastic resource provisioning such as storage, computation, and network bandwidth\\cite{yi2015fog}. \nDepending on their distance to IoT devices, fog nodes are often organized in a hierarchical fashion, with each layer as a \\textit{fog tier}.\nIn such a way, resource-limited IoT devices, when heavily loaded, can delegate workloads via wireless links to nearby fog nodes, \\textit{a.k.a.}, \\textit{workload offloading}, \nto reduce the power consumption and accelerate workload processing; \nmeanwhile, each fog node can offload workloads to nodes in its upper fog tier. \nHowever, along with all the benefits come the extended latency and extra power consumption. \nGiven such a power-latency tradeoff, two interesting questions arise.\nOne is \\textit{where} and \\textit{how much workloads} to offload between successive fog tiers. \nThe other is how to \\textit{allocate resources} for workload processing and offloading.\nThe timely decision making regarding these two questions is critical but challenging, due to temporal variations of system dynamics in wireless environment, uncertainty in the resulting offloading latency, and the unknown traffic statistics.\n\nWe summarize the main challenges of dynamic offloading and resource allocation in fog computing as follows: \n\\begin{enumerate}\n\t\\item[$\\diamond$] \\textbf{Characterization of system dynamics and the power-latency tradeoff}: In practice, a fog system often consists of multiple tiers, with complex interplays between fog tiers and the cloud, not to mention the constantly varying dynamics and intertwined power-latency tradeoffs therein.\n\t\tA model that accurately characterizes the system and tradeoffs is the key to the fundamental understanding of the design space.\n\t\\item[$\\diamond$] \\textbf{Efficient online decision making:}\n\t\tThe decision making must be computationally efficient, so as to minimize the overheads. The difficulties often come from the uncertainties of traffic statistics, online nature of workload arrivals, and intrinsic complexity of the problem.\n\t\\item[$\\diamond$] \\textbf{Understanding the benefits of predictive offloading:} One natural extension to online decision making is to employ predictive offloading to further reduce latencies and improve quality of service. For example, Netflix preloads videos onto users' devices based on user behavior prediction\\cite{NetflixPred}. Despite the wide applications of such approaches, the fundamental limits of predictive offloading in fog computing still remain unknown.\n\\end{enumerate}\n\n\n\\begin{table*}[!h]\n\\centering\n\\caption{Comparisons of related works}\n\\label{table: related works}\n\\begin{threeparttable}\t\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n & D2D-enabled IoT & IoT-Fog\\tnote{1} & Fog-Fog\\tnote{2} & Fog-Cloud\\tnote{3} & Dynamic & Prior Arrival Distribution & Prediction \\\\\n\\hline\n\\cite{xiao2017qoe} & & \\checkmark & \\checkmark & \\checkmark & & -- & \\\\\n\\hline\n\\cite{wang2019cooperative} & & \\checkmark & & \\checkmark & & -- & \\\\\n\\hline\n\\cite{liu2018socially} & & \\checkmark & & \\checkmark & \\checkmark & Poisson & \\\\\n\\hline \n\\cite{misra2019detour} & & \\checkmark & & & & -- & \\\\\n\\hline\n\\cite{lei2019joint} & & \\checkmark & & & \\checkmark & Poisson & \\\\\n\\hline\n\\cite{mao2016power} & & \\checkmark & & & \\checkmark & Not Required & \\\\\n\\hline\n\\cite{chen2019energy} & & \\checkmark & & & \\checkmark & Not Required & \\\\\n\\hline\n\\cite{gao2019dynamic} & \\checkmark & \\checkmark & & & \\checkmark & Not Required & \\\\\n\\hline\n\\cite{zhang2019near} & & \\checkmark & & & \\checkmark & Not Required & \\\\\n\\hline\nOurs & & & \\checkmark & \\checkmark & \\checkmark & Not Required & \\checkmark \\\\\n\\hline\n\\end{tabular}\n\\begin{tablenotes}\n\t\\footnotesize\n\t \\item[1,2,3] ``IoT-Fog'' means offloading from IoT devices to fog, ``Fog-Fog'' means offloading between fog tiers, while ``Fog-Cloud'' means offloading from fog to cloud. \n\\end{tablenotes}\n\\end{threeparttable}\n\t\\vspace{-0.5cm} \n\\end{table*}\n\nIn this paper, we focus on the workload offloading problem for multi-tiered fog systems. We address the above challenges by developing a fine-grained queueing model that accurately depicts such systems and proposing an efficient online scheme that proceeds the offloading on a time-slot basis. \nTo the best of our knowledge, we are the first to conduct systematic study on predictive offloading in fog systems. \nOur key results and main contributions are summarized as follows:\n\\begin{enumerate}\n\t\\item[$\\diamond$] \\textbf{Problem Formulation:} We formulate the problem of dynamic offloading and resource allocation as a stochastic optimization problem, aiming at minimizing the long-term time-average expectation of total power consumptions of fog tiers with queue stability guarantee.\n\t\\item[$\\diamond$] \\textbf{Algorithm Design:} \n\t\tThrough a non-trivial transformation, \n\t\twe decouple the problem into a series of subproblems over time slots. By exploiting their unique structures, we propose PORA, an efficient scheme that exploits predictive scheduling to make decisions in an online manner.\n\t\\item[$\\diamond$] \\textbf{Theoretical Analysis and Experimental Verification:} We conduct theoretical analysis and trace-driven simulations to evaluate the effectiveness of PORA. \n\t\tThe results show that PORA achieves a tunable power-latency tradeoff while effectively reducing the average latency with only mild-value of predictive information, even in the presence of prediction errors.\n\t\\item[$\\diamond$] \\textbf{New Degree of Freedom in the Design of Fog Computing Systems:} We systematically investigate the fundamental benefits of predictive offloading in fog computing systems, with both theoretical analysis and numerical evaluations.\n\\end{enumerate}\n\n\nWe organize the rest of the paper as follows. \nSection \\ref{sec: related work} discusses the related work. \nNext, in Section \\ref{sec: motivating example}, we provide an example that motivates our design for dynamic offloading and resource consumption in fog computing systems. \nSection \\ref{sec: model} presents the system model and problem formulation, followed by the algorithm design of PORA and performance analysis in Section \\ref{sec: algorithm}.\nSection \\ref{sec: simulation} analyzes the results from trace-driven simulations, while Section \\ref{sec: conclusion} concludes the paper. \n\n\n\\section{Related Work}\\label{sec: related work}\n\nIn recent years, a series of works have been proposed to optimize the performance fog computing systems from various aspects \\cite{xiao2017qoe, taneja2017resource, chen2019dynamic, wang2019cooperative, liu2018socially, misra2019detour, lei2019joint, mao2016power, chen2019energy, gao2019dynamic, zhang2019near}.\nAmong such works, the most related are those focusing on the design of effective offloading schemes. \nFor example, \nby adopting alternating direction method of multipliers (ADMM) methods,\nXiao \\textit{et al.}\\cite{xiao2017qoe} and Wang \\textit{et al.}\\cite{wang2019cooperative} proposed two offloading schemes for cloud-aided fog computing systems to minimize average task duration and average service response time under different energy constraints, respectively. \nLater, Liu \\textit{et al.} \\cite{liu2018socially} took the social relationships among IoT users into consideration and developed a socially aware offloading scheme by advocating game theoretic approaches. \nMisra \\textit{et al.} \\cite{misra2019detour} studied the problem in software-defined fog computing systems and proposed a greedy heuristic scheme to conduct multi-hop task offloading with offloading path selection. \nLei \\textit{et al.} \\cite{lei2019joint} considered the joint minimization of delay and power consumption over all IoT devices; they formulated the problem under the settings of continuous-time Markov decision process and solved it via approximate dynamic programming techniques.\nThe above works, despite their effectiveness, generally assume the availability of the statistical information on task arrivals in the systems which is usually unattainable in practice with highly time-varying system dynamics \\cite{zhang2017resource}.\n\n\n\nIn the face of such uncertainties, a number of works have applied stochastic optimization methods such as Lyapunov optimization techniques to online and dynamic offloading scheme design \n\\cite{mao2016power, chen2019energy, gao2019dynamic, zhang2019near}. \nFor instance, Mao \\textit{et al.}\\cite{mao2016power} investigated the tradeoff between the power consumption and execution delay, then developed a dynamic offloading scheme for energy-harvesting-enabled IoT devices.\nChen \\textit{et al.} \\cite{chen2019energy} designed an adaptive and efficient offloading scheme to minimize the transmission energy consumption with queueing latency guarantee.\nGao \\textit{et al.} \\cite{gao2019dynamic} investigated efficient offloading and social-awareness-aided network resource allocation for device-to-device-enabled (D2D-enabled) IoT users. \nZhang \\textit{et al.} \\cite{zhang2019near} designed an online rewards-optimal scheme for the computation offloading of energy harvesting-enabled IoT devices based on Lyapunov optimization and Vickrey-Clarke-Groves auction.\nDifferent from such works that focus on fog computing systems with flat or two-tiered architectures, \nour solution is applicable to general multi-tiered fog computing systems with time-varying wireless channel states and unknown traffic statistics. \nMoreover, to the best of our knowledge, \nour solution is also the first to proactively leverage the predicted traffic information to optimize the system performance with theoretical guarantee. \nWe are also the first to investigate the fundamental benefits of predictive offloading in fog computing systems.\nWe compare our work with the above mentioned works \nin TABLE \\ref{table: related works}.\n\n\n \n\\section{Motivating Example}\\label{sec: motivating example}\n\nIn this section, we provide a motivating example to show the potential power-latency tradeoff in multi-tiered fog computing systems. \nThe objective is to achieve low power consumptions and short average workload latency (in the unit of packets). \n\nFigure \\ref{figure: motivating example} shows an instance of time-slotted fog computing system with two fog tiers, \\textit{i.e.}, edge fog tier and central fog tier. \nWithin each fog tier resides one fog node, \\textit{i.e.}, an edge fog node (EFN) in edge fog tier and a central fog node (CFN) in central fog tier. \nThe EFN connects to the CFN via a wireless link, while the CFN connects to the cloud data center over wired links. \nEach fog node maintains one queue to store packets. \nFigure \\ref{figure: motivating example}(a) shows that during time slot $t_{0}$, both the EFN and the CFN store $8$ packets in their queues. \n\nWe assume that each fog node sticks to one policy all the time to handle packets, \\textit{i.e.}, either \\textit{processing packets locally} or \\textit{offloading them to its next tier}.\nThe local processing capacities of EFN and CFN are $1$ and $8$ packets per time slot, respectively.\nThe transmission capacities from EFN to CFN and from CFN to cloud are $4$ and $5$ packets per time slot, respectively. \nThe power consumption is assumed linearly proportional to the number of processed\/transmitted packets. \nIn particular, processing one packet locally consumes $1$ mW power, while transmitting one packet over wireless link consumes $0.5$ mW. \nWe ignore the processing latency in the cloud due to its powerful processing capacity. \n\nTABLE \\ref{table: motivating example} lists the total power consumptions and average packet latencies under all four possible settings. \nFigures \\ref{figure: motivating example}(b)-\\ref{figure: motivating example}(d) show the case when EFN sticks to offloading and CFN sticks to local processing. \nIn time slot $(t_{0}+1)$, EFN offloads four packets to CFN at its full transmission capacity, while CFN processes all the eight packets locally. \nIn time slot $(t_{0}+2)$, EFN offloads the rest four packets to CFN; meanwhile, CFN locally processes the four packets that arrive in previous time slot. \nIn time slot $(t_{0}+3)$, CFN finishes processing the rest four packets. \nIn this case, the system consumes $16$ mW power in local processing and $4$ mW power in transmission, with an average packet latency of $1.75$ time slots. \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[scale=0.44]{figures1\/motivatingExample}\n\t\\caption{Motivating example of dynamic offloading and resource consumption in multi-tiered fog computing systems.}\n\t\\label{figure: motivating example}\n\\end{figure}\n\n\\begin{table}[!h]\n\\centering\n\\caption{Performance under different offloading policies}\n\\label{table: motivating example}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nPolicy of & Policy of & Total Power & Average Packet \\\\ \nEFN & CFN & Consumptions (mW) & Latency (time slot) \\\\ \\hline\nLocal & Local & 16 & 2.75 \\\\ \\hline\nLocal & Offload & 8 & 2.9375 \\\\ \\hline\nOffload & Local & 20 & 1.75 \\\\ \\hline\nOffload & Offload & 4 & 2.125 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nFrom TABLE \\ref{table: motivating example}, we conclude that: \nFirst, when EFN sticks to offloading and CFN sticks to local processing, the system achieves the lowest average packet latency of $1.75$ slots but the maximum power consumption of $20$mW. \nSecond, with the same offloading policy on EFN, there is a tradeoff between the total power consumptions and the average packet latency when CFN sticks to different policies. \nThe reason is that offloading to the cloud can not only reduce power consumptions but also prolong latency as well. \nThird, when CFN sticks to local processing, there is a power-latency tradeoff with different policies at EFN, in that offloading to CFN can induce lower processing latency but at the cost of even higher power consumption for wireless transmissions.\n\n\\section{Model and Problem Formulation}\n\\label{sec: model}\n\nWe consider a multi-tiered fog computing system, as shown in Figure \\ref{figure: hierarchical fog system}.\nThe system evolves over time slots indexed by $t \\in \\{0, 1, 2,...\\}$.\nEach time slot has a length of $\\tau_{0}$.\nInside the edge fog tier (EFT) are a set of edge fog nodes (EFNs) that offer low-latency access to IoT devices.\nOn the other hand, the central fog tier (CFT) comprises of central fog nodes (CFNs) with greater processing capacities than EFNs.\nWe assume that the workload on each EFN can be offloaded to and processed by any of its accessible CFNs, and that each CFN can offload its workload to the cloud.\nIn our model, we do not consider the power consumptions and latencies within the cloud. We mainly focus on the power consumptions and latencies within fog tiers, as shown in TABLE \\ref{table: model}. First, the power consumptions we consider include two parts: processing power and transmit power. The processing power consumption is induced by the workload processing on both EFT and CFT. The transmit power is induced by the transmissions from EFT to CFT. We do not consider the transmit power consumption from CFT to cloud because we assume that the CFT communicates with the cloud through wireline connections. Second, the latencies we consider include three parts: queueing latency, processing latency and transmit latency. We focus on the queueing latency on both EFT and CFT. We assume that the workload processing in each time slot can be completed by the end of the same time slot, and then we can ignore the processing latency. Since the EFT communicates with the CFT through high-speed wireless connections and the CFT communicates with the cloud through high-speed wireline connections, we assume that transmission latencies from both EFT to CFT and CFT to Cloud are negligible.\n\n\\begin{table}[!h]\n\\centering\n\\caption{Performance Metrics in Our Model}\n\\label{table: model}\n\\begin{tabular}{|p{1.2cm}<{\\centering}|p{1.1cm}<{\\centering}|p{0.9cm}<{\\centering}|p{1.0cm}<{\\centering}|p{1.1cm}<{\\centering}|p{0.9cm}<{\\centering}|}\n\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c|}{Power Consumption} & \\multicolumn{3}{c|}{Latency} \\\\ \\cline{2-6} \n& Processing & Transmit & Queueing & Processing & Transmit \\\\ \\hline\nEFT & \\checkmark & & \\checkmark & & \\\\ \\hline\nEFT2CFT & & \\checkmark & & & \\\\ \\hline\nCFT & \\checkmark & & \\checkmark & & \\\\ \\hline\nCFT2Cloud & & & & & \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\nIn the following, we first introduce the basic settings in Section \\ref{subsec: basic setting}, then elaborate the queueing models in Section \\ref{subsec: local queue}. \nNext, we define the optimization objective in Section \\ref{subsec: power} while proposing the problem formulation in Section \\ref{subsec: formulation}.\nWe summarize the key notations in TABLE \\ref{table: key notations}.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[scale=0.35]{figures1\/edgeCentralFog}\n\t\\caption{An example of fog computing systems with two fog tiers.}\n\t\\label{figure: hierarchical fog system}\n\\end{figure}\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Key notations}\n\\label{table: key notations}\n\\centering\n\\begin{tabular}{p{0.89cm} l}\n \\hline\\hline\n Notation & Description \\\\ \\hline\n $\\tau_{0}$ & Length of each time slot \\\\ \\hline\n $\\mathcal{N}$ & $\\mathcal{N}$ is the set of EFNs with $|\\mathcal{N}|\\triangleq N$ \\\\ \\hline\n $\\mathcal{M}$ & $\\mathcal{M}$ is the set of CFNs with $|\\mathcal{M}|\\triangleq M$ \\\\ \\hline\n $\\mathcal{N}_{j}$ & Set of accessible EFNs from CFN $j$ \\\\ \\hline \n $\\mathcal{M}_{i}$ & Set of accessible CFNs from EFN $i$ \\\\ \\hline\n\t$A_{i}(t)$ & Amount of workload arriving to EFN $i$ in time slot $t$ \\\\ \\hline\n\t$\\lambda_{i}$ & Average workload arriving rate on EFN $i$, $\\lambda_{i}\\triangleq \\mathbb{E}\\{A_{i}(t)\\}$ \\\\ \\hline\n\t$W_{i}$ & Prediction window size of EFN $i$ \\\\ \\hline\n\t$A_{i,-1}(t)$ & Arrival queue backlog of EFN $i$ in time slot $t$ \\\\ \\hline\n\t\\multirow{2}*{$A_{i,w}(t)$} & Prediction queue backlog of EFN $i$ in time slot $t$, such that \\\\ \n\t& $0\\leq w\\leq W_{i}-1$ \\\\ \\hline\n\t$Q_{i}^{(e,a)}(t)$ & Integrate queue backlog of EFN $i$ in time slot $t$ \\\\ \\hline\n\t$Q_{i}^{(e,l)}(t)$ & Local processing queue backlog of EFN $i$ in time slot $t$ \\\\ \\hline\n\t$Q_{i}^{(e,o)}(t)$ & Offloading queue backlog of EFN $i$ in time slot $t$ \\\\ \\hline\n\t$b_{i}^{(e,l)}(t)$ & Amount of workload to be sent to $Q_{i}^{(e,l)}(t)$ in time slot $t$ \\\\ \\hline\n\t$b_{i}^{(e,o)}(t)$ & Amount of workload to be sent to $Q_{i}^{(e,o)}(t)$ in time slot $t$ \\\\ \\hline\n\t$f_{i}^{(e)}(t)$ & CPU frequency of EFN $i$ in time slot $t$ \\\\ \\hline\n\t$H_{i,j}(t)$ & Wireless channel gain between EFN $i$ and CFN $j$ \\\\ \\hline\n\t$p_{i,j}(t)$ & Transmit power from EFN $i$ to CFN $j$ in time slot $t$ \\\\ \\hline\n\t$R_{i,j}(t)$ & Transmit rate from EFN $i$ to CFN $j$ in time slot $t$ \\\\ \\hline\n\t$Q_{j}^{(c,a)}(t)$ & Arrival queue backlog of CFN $j$ in time slot $t$ \\\\ \\hline\n\t$Q_{j}^{(c,l)}(t)$ & Local processing queue backlog of CFN $j$ in time slot $t$ \\\\ \\hline\n\t$Q_{j}^{(c,o)}(t)$ & Offloading queue backlog of CFN $j$ in time slot $t$ \\\\ \\hline\n\t$b_{j}^{(c,l)}(t)$ & Amount of workload to be sent to $Q_{j}^{(c,l)}(t)$ in time slot $t$ \\\\ \\hline\n\t$b_{j}^{(c,o)}(t)$ & Amount of workload to be sent to $Q_{j}^{(c,o)}(t)$ in time slot $t$ \\\\ \\hline\n\t$f_{j}^{(c)}(t)$ & CPU frequency of CFN $j$ in time slot $t$ \\\\ \\hline\n\t$P(t)$ & Total power consumptions in time slot $t$ \\\\ \\hline\n\t\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Basic Settings}\\label{subsec: basic setting}\n\nThe fog computing system consists of $N$ EFNs in EFT and $M$ CFNs in CFT. \nLet $\\mathcal{N}$ and $\\mathcal{M}$ be the sets of EFNs and CFNs.\nEach EFN $i$ has access to a subset of CFNs in their proximities. We denote the subset by $\\mathcal{M}_{i}\\subset\\mathcal{M}$. \nFor each CFN $j$, $\\mathcal{N}_{j}\\subset\\mathcal{N}$ denotes the set of its accessible EFNs. \nAccordingly, for any $i\\in\\mathcal{N}_{j}$ we have $j\\in\\mathcal{M}_{i}$. \n\n\\subsection{Queueing Model for Edge Fog Node}\\label{subsec: queue for EFN}\n\nDuring time slot $t$, there is an amount $A_{i}(t)$ ($\\le A_{\\text{max}}$ for some constant $A_{\\text{max}}$) of workload generated from IoT devices arrive to be processed on EFN $i$ such that $\\mathbb{E}\\{ A_{i}(t)\\} =\\lambda_{i}$.\nWe assume that such arrivals are independent over time slots and different EFNs. \nEach EFN $i$ is equipped with a learning module\\footnote{We do not specify any particular learning method in this paper, since our work aims to explore the \\textit{fundamental} benefits of predictive offloading. \nIn practice, one can leverage machine learning techniques such as time-series prediction methods \\cite{ahmed2010empirical} for workload arrival prediction.}\nthat can predict the future workload within a \\textit{prediction window} of size $W_{i}$, \\textit{i.e.} workload will arrive in the next $W_{i}$ time slots. The predicted arrivals are pre-generated and recorded, then arrive to EFN $i$ for pre-serving. Once the predicted arrivals actually arrive after pre-serving, they will be considered finished.\n\nOn each EFN, as Figure \\ref{figure: queue} shows, there are four types of queues:\nprediction queues with the backlogs as $A_{i,0}(t)$, ..., $A_{i,W_{i}-1}(t)$, \narrival queue $A_{i,-1}(t)$, \nlocal processing queue $Q_{i}^{(e,l)}(t)$, and offloading queue $Q_{i}^{(e,o)}(t)$.\nIn time slot $t$, prediction queue $A_{i,w}(t)$ ($0\\leq w\\leq W_{i}-1$) stores untreated workload that will arrive in time slot $(t+w)$. \nWorkload that actually arrives at EFN $i$ is stored in the arrival queue $A_{i,-1}(t)$, \nawaiting being forwarded to the local processing queue $Q_{i}^{(e,l)}(t)$ or the offloading queue $Q_{i}^{(e,o)}(t)$. \nWorkload in $Q_{i}^{(e,l)}(t)$ will be processed locally by EFN $i$, \nwhile workload in $Q_{i}^{(e,o)}(t)$ will be offloaded to CFNs in set $\\mathcal{M}_{i}$.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[scale=0.3]{figures1\/model}\n\t\\caption{Queueing model of the system.}\n\t\\label{figure: queue}\n\\end{figure}\n\n\\subsubsection{Prediction Queues and Arrival Queues in EFNs}\n\nWithin each time slot $t$, in addition to the current arrivals in the arrival queue, EFN $i$ can also forward future arrivals in the prediction queues.\nWe define $\\mu_{i,w}(t)$ as the amount of output workload from $A_{i,w}(t)$, for $w\\in\\{-1,0,...,W_{i}-1\\}$.\nSuch workload should be distributed to the local processing queue and offloading queue.\nWe denote the amounts of workloads to be distributed to the local processing queue and offloading queue as $b_{i}^{(e,l)}(t)$ and $b_{i}^{(e,o)}(t)$, respectively, such that\n\\begin{equation}\n\t0\\leq b^{(e,\\beta)}_{i}\\left(t\\right)\\leq b^{(e,\\beta)}_{i,\\text{max}},\\ \\forall \\beta\\in\\{l,o\\} \\label{constraint: EFN offloading decision 1}\n\\end{equation}\nwhere each $b^{(e,\\beta)}_{i,\\text{max}}$ is a positive constant.\nAs a result, we have\n\\begin{equation}\\label{constraint: pre-serve rates}\n\t\\sum_{w=-1}^{W_{i}-1}\\mu_{i,w}\\left(t\\right)= b^{(e,l)}_{i}\\left(t\\right)+b^{(e,o)}_{i}\\left(t\\right).\n\\end{equation}\n\nNext, we consider the queueing dynamics for different types of queues in EFN, respectively.\n\nRegarding $A_{i,w}(t)$, it is updated whenever pre-service is finished and the lookahead window moves one slot ahead at the end of each time slot. \nTherefore, we have\n\\begin{enumerate}[(i)]\n\t\\item If $w=W_{i}-1$, then\n\t\\begin{equation}\\label{update: fog node prediction queue update 1}\n\t\tA_{i,W_{i}-1}\\left(t+1\\right)=A_{i}\\left(t+W_{i}\\right).\n\t\\end{equation}\n\t\\item If $0\\leq w\\leq W_{i}-2$, then\n\t\\begin{equation}\\label{update: fog node prediction queue update 2}\n\t\tA_{i,w}(t+1)=[A_{i,w+1}(t)-\\mu_{i,w+1}(t)]^{+},\n\t\\end{equation}\n\\end{enumerate}\nwhere $[x]^{+}\\triangleq \\max\\{x,0\\}$ for $x \\in \\mathbb{R}$. \nIn time slot $(t+1)$, the amount of workload that will arrive after $(W_{i}-1)$ time slots is $A_{i}(t+W_{i})$ and it remains unknown until time slot $(t+1)$. \n\nRegarding the arrival queue $A_{i,-1}(t)$, it records the actual backlog of EFN $i$ with the update equation as follows:\n\\begin{equation}\\label{update: fog node true queue update}\n\tA_{i,-1}(t+1)\\!=\\![A_{i,-1}(t)-\\mu_{i,-1}(t)]^{+}\\!+\\![A_{i,0}(t)-\\mu_{i,0}(t)]^{+}.\n\\end{equation}\nNote that $\\mu_{i,-1}(t)$ denotes the amount of distributed workload that have already being in $A_{i,-1}(t)$. \n\nNext, we introduce an integrate queue with a backlog size as the sum of all prediction queues and the arrival queue on EFN $i$, denoted by $Q_{i}^{\\left(e,a\\right)}\\left(t\\right)\\triangleq \\sum_{w=-1}^{W_{i}-1}A_{i,w}\\left(t\\right)$ . \nUnder \\textit{fully-efficient} \\cite{huang2016predictive} service policy, $Q_{i}^{\\left(e,a\\right)}\\left(t\\right)$ is updated as\n\\begin{multline}\\label{update: prediction sum queue update}\n\tQ_{i}^{\\left(e,a\\right)}\\left(t+1\\right)=[Q_{i}^{\\left(e,a\\right)}\\left(t\\right)-(b^{(e,l)}_{i}\\left(t\\right)+b^{(e,o)}_{i}\\left(t\\right))]^{+}\\\\\n\t+A_{i}\\left(t+W_{i}\\right).\n\\end{multline}\nThe input of integrate queue $Q_{i}^{(e,a)}(t)$ consists of the predicted workload that will arrive at EFN $i$ in time slot $(t+W_{i})$, while its output consists of workloads being forwarded to the local processing queue and the offloading queue. \nNote that $b^{(e,l)}_{i}(t)+b^{(e,o)}_{i}(t)$ is the output capacity of integrate queue $Q_{i}^{(e,a)}(t)$ in time slot $t$. If the capacity is larger than the queue backlog size, the true output amount will be smaller than $b^{(e,l)}_{i}(t)+b^{(e,o)}_{i}(t)$.\n \n\n\\subsubsection{Offloading Queues in EFNs}\n\nIn time slot $t$, workload in queue $Q^{(e,o)}_{i}(t)$ will be offloaded to CFNs in set $\\mathcal{M}_{i}$. \nThe transmission capacities are determined by the transmit power decisions $(p_{i,j}(t))_{j\\in\\mathcal{M}_{i}}$, where $p_{i,j}(t)$ is the transmit power from EFN $i$ to CFN $j$. \nThe transmit power is nonnegative and the total transmit power of each EFN is upper bounded, \\textit{i.e.}, \n\\begin{align}\n\t&p_{i,j}\\left(t\\right)\\geq0,\\ \\forall i\\in\\mathcal{N},j\\in\\mathcal{M}_{i}\\text{ and }t, \\label{constraint: power allocation 1} \\\\\n\t&\\sum_{j\\in\\mathcal{M}_{i} }p_{i,j}\\left(t\\right)\\leq p_{i,\\text{max}},\\ \\forall i\\in\\mathcal{N}\\text{ and }t.\\label{constraint: power allocation 2}\n\\end{align}\nAccording to Shannon's capacity formula \\cite{gallager2008principles}, the transmission capacity from EFN $i$ to CFN $j$ is\n\\begin{equation}\\label{equation: offload rate to central fog}\n\tR_{i,j}(t)\\!\\triangleq\\!\\hat{R}_{i,j}(p_{i,j}(t))\\!=\\!\\tau_{0}B\\log_{2}\\left(1\\!+\\!\\frac{p_{i,j}(t)H_{i,j}(t)}{N_{0}B}\\right),\n\\end{equation}\nwhere $\\tau_{0}$ is the length of each time slot, $B$ is the channel bandwidth, $H_{i,j}(t)$ is the wireless channel gain between EFN $i$ and CFN $j$, and $N_{0}$ is the system power spectral density of the additive white Gaussian noise.\nNote that $H_{i,j}(t)$ is an uncontrollable environment state with positive upper bound $H_{\\text{max}}$.\nWe do not consider the interferences among fog nodes and tiers. \nBy adjusting the transmit power $p_{i,j}(t)$, we can offload different amounts of workload from EFN $i$ to CFN $j$ in time slot $t$. \nAccordingly, the update equation of offloading queue $Q^{(e,o)}_{i}(t)$ is\n\\begin{equation}\\label{update: edge offloading queue update}\n\tQ^{(e,o)}_{i}\\left(t+1\\right)\\leq [Q^{(e,o)}_{i}\\left(t\\right)\\!-\\!\\sum_{j\\in\\mathcal{M}_{i}}R_{i,j}\\left(t\\right)]^{+}+b^{(e,o)}_{i}\\left(t\\right),\n\\end{equation}\nwhere $\\sum_{j \\in \\mathcal{M}_{i}} R_{i,j}(t)$ is the total transmission capacity to EFN $i$ in time slot $t$.\nThe inequality here means that the actual arrival of $Q_{i}^{(e,o)}(t)$ may be less than $b_{i}^{(e,o)}(t)$, because $b^{(e,o)}_{i}(t)$ is the transmission capacity from integrate queue $Q_{i}^{(e,a)}(t)$ to offloading queue $Q_{i}^{(e,o)}(t)$ instead of the amount of truly transmitted workload. \nRecall that we assume the transmission latency from EFT to CFT is negligible compared to the length of each time slot, the workload transmission in each time slot can be accomplished by the end of that time slot.\n\n\\subsection{Queueing Model for Central Fog Node}\\label{subsec: queue for CFN}\n\nFigure \\ref{figure: queue} also shows the queueing model on CFN.\nEach CFN $j\\in\\mathcal{M}$ maintains three queues: \nan arrival queue $Q_{j}^{(c,a)}(t)$, a local processing queue $Q_{j}^{(c,l)}(t)$, and an offloading queue $Q_{j}^{(c,o)}(t)$. Similar to EFNs, workload offloaded from the EFT will be firstly stored in the arrival queue,\nthen distributed to $Q_{j}^{(c,l)}(t)$ for local processing and to $Q_{j}^{(c,o)}(t)$ for further offloading.\n\n\n\\subsubsection{Arrival Queues in CFNs}\n\nThe arrivals on CFN $j$ consist of \nworkloads offloaded from EFNs in the set $\\mathcal{N}_{j}$. \nWe denote the amounts of workloads distributed to the local processing queue and offloading queue in time slot $t$ as $b_{j}^{(c,l)}(t)$ and $b_{j}^{(c,o)}(t)$, respectively, such that\n\\begin{equation}\n\t0\\leq b^{(c,\\beta)}_{j}\\left(t\\right)\\leq b^{(c,\\beta)}_{j,\\text{max}},\\ \\forall \\beta\\in\\{l,o\\}, \\label{constraint: CFN offloading decision 1} \n\\end{equation}\nwhere each $b^{(c,\\beta)}_{j,\\text{max}}$ is a positive constant. Accordingly, $Q_{j}^{(c,a)}(t)$ is updated as follows:\n\\begin{multline}\\label{update: central arrival queue update}\n\tQ_{j}^{(c,a)}(t+1)\\\\\n\t\\leq [Q_{j}^{(c,a)}(t)-(b_{j}^{(c,l)}(t)+b_{j}^{(c,o)}(t))]^{+}\\!+\\!\\sum_{i\\in\\mathcal{N}_{j}}R_{i,j}(t).\n\\end{multline}\n\n\\subsubsection{Offloading Queues in CFNs}\n\nFor each CFN $j\\in\\mathcal{M}$, its offloading queue $Q^{(c,o)}_{j}(t)$ stores the workload to be offloaded to the cloud. \nWe define $D_{j}(t)$ as the transmission capacity of the wired link from CFN $j$ to the cloud during time slot $t$, \nwhich depends on the network state and is upper bounded by some constant $D_{\\text{max}}$ for all $j$ and $t$. Then we have the following update function for $Q^{(c,o)}_{j}(t)$:\n\\begin{equation}\\label{update: central offloading queue update}\n\tQ^{(c,o)}_{j}\\left(t+1\\right)\\leq [Q^{(c,o)}_{j}\\left(t\\right)-D_{j}\\left(t\\right)]^{+}+b^{(c,o)}_{j}\\left(t\\right).\n\\end{equation}\nNote that the amount of actually offloaded workload to the cloud is $\\min\\{Q_{j}^{(c,o)}(t),D_{j}(t)\\}$.\n\n\\subsection{Local Processing Queues on EFNs and CFNs}\\label{subsec: local queue}\n\nWe assume that all fog nodes are able to adjust their CPU frequencies in each time slot, by applying \\textit{dynamic voltage and frequency scaling} (DVFS) techniques\\cite{mao2017survey}. \nNext, we define $L_{k}^{(\\alpha)}$ as the number of CPU cycles that fog node $k\\in\\mathcal{N}\\cup\\mathcal{M}$ requires to process one bit of workload, where $\\alpha$ is an indicator of fog node $k$'s type ($\\alpha=e$ if $k$ is an EFN, and $\\alpha=c$ if $k$ is a CFN). \n$L_{k}^{(\\alpha)}$ is assumed constant and can be measured offline \\cite{miettinen2010energy}. \nTherefore, the local processing capacity of fog node $k$ is $f_{k}^{(\\alpha)}(t)\/L_{k}^{(\\alpha)}$.\nThe local processing queue on fog node $k$ evolves as follows:\n\\begin{equation}\\label{update: local processing queue update}\n\tQ_{k}^{(\\alpha,l)}(t+1)\\!\\leq \\![Q_{k}^{\\left(\\alpha,l\\right)}\\left(t\\right)\\!-\\tau_{0}f_{k}^{\\left(\\alpha\\right)}\\left(t\\right)\\!\/\\!L_{k}^{\\left(\\alpha\\right)}]^{+}\\!+b_{k}^{\\left(\\alpha,l\\right)}\\left(t\\right).\n\\end{equation}\nAll CPU frequencies are nonnegative and finite:\n\\begin{equation}\\label{constraint: CPU frequency}\n\t0\\leq f_{k}^{(\\alpha)}\\left(t\\right)\\leq f^{(\\alpha)}_{k,\\text{max}},\\ \\forall k\\in\\mathcal{N}\\cup\\mathcal{M}\\text{ and }t,\n\\end{equation}\nwhere each $f^{(\\alpha)}_{k,\\text{max}}$ is a positive constant. \n\n\\subsection{Power Consumptions}\\label{subsec: power}\n\nThe total power consumptions $P(t)$ of fog tiers in time slot $t$\nconsist of the processing power consumption and wireless transmit power consumption. \nGiven a local CPU with frequency $f$, its power consumption per time slot is $\\tau_{0}\\varsigma f^{3}$, where $\\varsigma$ is a parameter depending on the deployed hardware and is measurable in practice \\cite{kim2018dual}. \nThus $P(t)$ is defined as follows:\n\\begin{multline}\\label{equation: total fog power consumptions}\n\tP\\left(t\\right)\\triangleq\\hat{P}\\left(\\boldsymbol{f}\\left(t\\right),\\boldsymbol{p}\\left(t\\right)\\right)=\\sum_{i\\in\\mathcal{N}}\\tau_{0}\\varsigma(f_{i}^{\\left(e\\right)}\\left(t\\right))^{3}\\\\\n\t+\\sum_{j\\in\\mathcal{M}}\\tau_{0}\\varsigma(f_{j}^{\\left(c\\right)}\\left(t\\right))^{3}+\\sum_{i\\in\\mathcal{N}}\\sum_{j\\in\\mathcal{M}_{i}} \\tau_{0}p_{i,j}\\left(t\\right),\n\\end{multline}\nwhere $\\boldsymbol{f}(t)\\triangleq((f_{i}^{(e)}(t))_{i\\in\\mathcal{N}},(f_{j}^{(c)}(t))_{j\\in\\mathcal{M}})$ is the vector of all CPU frequencies, and $\\boldsymbol{p}(t)\\triangleq(\\boldsymbol{p}_{i}(t))_{i\\in\\mathcal{N}}$ in which $\\boldsymbol{p}_{i}(t)=(p_{i,j}(t))_{j\\in\\mathcal{M}_{i}}$ is the transmit power allocation of EFN $i$. \n\n\n\\subsection{Problem Formulation}\\label{subsec: formulation}\n\nWe define the long-term time-average expectation of total power consumptions $\\bar{P}\n$ and total queue backlog $\\bar{Q}$ as follows:\n\\begin{align}\n\t&~~~~~~~~~~~~~~\\bar{P}\\triangleq\\limsup_{T\\rightarrow\\infty}\\frac{1}{T}\\sum_{t=0}^{T-1}\\mathbb{E}\\left\\{ P\\left(t\\right)\\right\\}, \\label{definition: time average exp power}\\\\\n\t&\\bar{Q}\\triangleq\\limsup_{T\\rightarrow\\infty}\\frac{1}{T}\\sum_{t=0}^{T-1}\\sum_{\\beta\\in\\{ a,l,o\\} }(\\sum_{i\\in\\mathcal{N}}\\mathbb{E}\\{ Q_{i}^{\\left(e,\\beta\\right)}(t)\\}\\nonumber \\\\\n\t&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\\sum_{j\\in\\mathcal{M}}\\mathbb{E}\\{ Q_{j}^{(c,\\beta)}(t)\\}) \\label{definition: time average exp backlog}.\n\\end{align}\nIn this paper, we aim to minimize the long-term time-average expectation of total power consumptions $\\bar{P}$, while ensuring the stability of all queues in the system, \\textit{i.e.}, $\\bar{Q}<\\infty$. \nThe problem formulation is given by\n\\begin{equation}\\label{problem: general}\n\\begin{array}{cl}\n\\underset{\\{\\boldsymbol{b}(t),\\boldsymbol{f}(t),\\boldsymbol{p}(t)\\}_{t}}{\\text{Minimize}}\n\t&\\displaystyle \\bar{P}\\\\\n\t\\text{Subject to}&\\displaystyle (\\ref{constraint: EFN offloading decision 1})\n\t(\\ref{constraint: power allocation 1})\n\t(\\ref{constraint: power allocation 2})(\\ref{constraint: CFN offloading decision 1})(\\ref{constraint: CPU frequency}),\\\\\n\t&\\displaystyle \\bar{Q}<\\infty.\n\\end{array}\n\\end{equation}\n\n\n\\section{Algorithm Design}\\label{sec: algorithm}\n\n\\subsection{Predictive Algorithm}\n\nTo solve problem (\\ref{problem: general}), we adopt Lyapunov optimization techniques\\cite{huang2016predictive}\\cite{neely2010stochastic} to decouple the problem into a series of subproblems over time slots. We show the detail of this process in Appendix \\ref{appendix: pora design}. \nBy solving each of these subproblems during each time slot, \nwe propose PORA, an efficient and predictive scheme conducts workload offloading in an online and distributed manner. \nWe show the pseudocode of PORA in Algorithm \\ref{algorithm: pora}. \nNote that symbol $\\alpha \\in \\{ e, c \\}$ indicates the type of fog node. \nSpecifically, for each fog node $k$, $\\alpha=e$ if $k$ is an EFN and CFN otherwise.\n\\begin{algorithm}[t]\n\\caption{Predictive Offloading and Resource Allocation (PORA) in One Time Slot}\n\\label{algorithm: pora}\n\\begin{algorithmic}[1]\n \\STATE Initialize $\\boldsymbol{b}(t)\\leftarrow \\boldsymbol{0}$, $\\boldsymbol{f}(t)\\leftarrow\\boldsymbol{0}$, $\\boldsymbol{p}(t)\\leftarrow\\boldsymbol{0}$.\n \\FOR {each fog node $k\\in\\mathcal{N}\\cup\\mathcal{M}$}\n \\STATE \\%\\%\\textit{Make Offloading Decisions}\n \\IF {$Q_{k}^{(\\alpha,a)}(t)> Q_{k}^{(\\alpha,l)}(t)$}\n \\STATE Set $b_{k}^{(\\alpha,l)}(t)\\leftarrow b^{(\\alpha,l)}_{k,\\text{max}}$.\n \\ENDIF\n \\IF {$Q_{k}^{(\\alpha,a)}(t)> Q_{k}^{(\\alpha,o)}(t)$}\n \\STATE Set $b_{k}^{(\\alpha,o)}(t)\\leftarrow b^{(\\alpha,o)}_{k,\\text{max}}$. \n \\ENDIF\n \\STATE \\%\\%\\textit{Local CPU Resource Allocation}\n \\STATE Set $ f^{(\\alpha)}_{k}\\left(t\\right)\\leftarrow\\min\\{ \\sqrt{Q_{k}^{(\\alpha,l)}(t)\/3V\\varsigma L_{k}^{(\\alpha)}},f^{(\\alpha)}_{k,\\text{max}}\\}$.\n \\ENDFOR\n \\STATE \\%\\%\\textit{Transmit Power Allocation}\n \\FOR {each EFN $i\\in\\mathcal{N}$}\n \\STATE Set $\\lambda_{\\text{min}}\\leftarrow 0$.\n \\STATE Set $\\lambda_{\\text{max}}\\leftarrow \\max_{j\\in\\mathcal{M}_{i}}\\frac{(Q_{i}^{(e,o)}-Q_{j}^{(c,a)})H_{i,j}(t)}{N_{0}}-V$.\n\\WHILE{$\\lambda_{\\text{max}}-\\lambda_{\\text{min}}> \\varepsilon$}\n \\STATE \\%\\%\\textit{Water Filling with Bisection Method}\n \\STATE Set $\\lambda^{*}\\leftarrow (\\lambda_{\\text{min}}+\\lambda_{\\text{max}})\/2$ \n \\STATE Set $p_{i,j}(t)\\!\\leftarrow\\!B\\left[\\frac{Q_{i}^{(e,o)}(t)-Q_{j}^{(c,a)}(t)}{V+\\lambda^{*}}-\\frac{N_{0}}{H_{i,j}(t)}\\right]^{+}$.\n \\IF {$\\sum_{j\\in\\mathcal{M}_{i}}p_{i,j}(t)> p_{i,\\text{max}}$}\n \\STATE Set $\\lambda_{\\text{max}}\\leftarrow\\lambda^{*}$.\n \\ELSE\n \\STATE Set $\\lambda_{\\text{min}}\\leftarrow\\lambda^{*}$.\n \\ENDIF\n\\ENDWHILE\n\\ENDFOR\n \\STATE Enforce scheduling decisions $\\boldsymbol{b}(t)$, $\\boldsymbol{f}(t)$, and $\\boldsymbol{p}(t)$.\n\\end{algorithmic}\n\\end{algorithm}\nNext, we introduce PORA in detail.\n\n\n\\subsubsection{Offloading Decision}\n\nIn each time slot $t$, under PORA, each fog node $k\\in\\mathcal{N}\\cup\\mathcal{M}$ decides the amounts of workload scheduled to the offloading queue and the local processing queue, denoted by $b_{k}^{(\\alpha,l)}(t)$ and $b_{k}^{(\\alpha,o)}(t)$, respectively. \nSuch decisions are obtained by solving the following problem:\n\\begin{equation}\\label{problem: offloading decision}\n\\begin{array}{cl}\n\\underset{0\\leq b_{k}^{(\\alpha,\\beta)}\\leq b^{(\\alpha,\\beta)}_{k,\\text{max}}}{\\text{Minimize}}\n\t\\displaystyle \\left(Q^{(\\alpha,\\beta)}_{k}\\left(t\\right)-Q_{k}^{(\\alpha,a)}\\left(t\\right)\\right)b_{k}^{(\\alpha,\\beta)},\n\\end{array}\n\\end{equation}\nwhere $\\beta\\in\\{l,o\\}$. \nAccordingly, the optimal solution to (\\ref{problem: offloading decision}) is\n\\begin{equation}\\label{equation: optimal offload decision}\n\tb_{k}^{(\\alpha,\\beta)}\\left(t\\right)=\\begin{cases}\nb^{(\\alpha,\\beta)}_{k,\\text{max}}, & \\text{if }Q_{k}^{(\\alpha,\\beta)}\\left(t\\right)0$ and $\\epsilon>0$ such that\n\\begin{equation*}\n\t\\bar{P}\\leq \\theta\/V+P^{*},\\ \\bar{Q}\\leq (\\theta+VP_{\\text{max}})\/\\epsilon,\n\\end{equation*}\nwhere $\\bar{P}$ and $\\bar{Q}$ are defined in (\\ref{definition: time average exp power}) and (\\ref{definition: time average exp backlog}), respectively.}\n\\end{theorem}\nThe proof is quite standard and hence omitted here.\n\n\\textbf{Remark:}\nBy \\textit{Little's} theorem\\cite{leon2017probability}, the average queue backlog size is proportional to the average queueing latency. \nTherefore, Theorem \\ref{theorem: performance} implies that by adjusting parameter $V$, PORA can achieve an $[O(1\/V),O(V)]$ power-latency tradeoff in the non-predictive case. \nFurthermore, the average power consumption $\\bar{P}$ approaches the optimum $P^{*}$ asymptotically as the value of $V$ increases to infinity. \n\n\\subsubsection{Latency Reduction}\nWe analyze the latency reduction induced by PORA under perfect prediction compared to the non-predictive case. \nIn particular, we denote the prediction window vector $(W_{i})_{i\\in\\mathcal{N}}$ by $\\boldsymbol{W}$ and the corresponding delay reduction by $\\eta(\\boldsymbol{W})$. \nFor each unit of workload on EFN $i$, let $\\pi_{i,w}$ denote the steady-state probability that it experiences a latency of $w$ time slots in $A_{i,-1}(t)$. \nWithout prediction, \nthe average latency on its \\textit{arrival queues} is $\n\td=\\sum_{i\\in\\mathcal{N}}\\lambda_{i}\\sum_{w\\geq1}w\\pi_{i,w}\/\\sum_{i\\in\\mathcal{N}}\\lambda_{i}$. Then we have the following theorem.\n\\begin{theorem}\\label{theorem: delay}\n\\textit{Suppose the system steady-state behavior depends only on the statistical behaviors of the arrivals and service processes. Then the latency reduction $\\eta(\\boldsymbol{W})$ is\n\\begin{multline}\\label{theorem 2: result 1}\n\t\\eta\\left(\\boldsymbol{W}\\right)\\\\\n\t=\\frac{\\sum_{i\\in\\mathcal{N}}\\lambda_{i}\\!\\left(\\!\\sum_{1\\leq w\\leq W_{i}}\\!w\\pi_{i,w}\\!+\\!W_{i}\\!\\sum_{w\\geq1}\\!\\pi_{i, w+W_{i}}\\right)}{\\sum_{i\\in\\mathcal{N}}\\lambda_{i}}.\n\\end{multline}\nFurthermore, if $d<\\infty$, as $\\boldsymbol{W}\\rightarrow\\infty$, \\textit{i.e.}, \nwith inifinite predictive information, we have\n\\begin{equation}\\label{theorem 2: result 2}\n\t\\lim_{\\boldsymbol{W}\\rightarrow\\infty}\\eta\\left(\\boldsymbol{W}\\right)=d.\n\\end{equation}\n}\n\\end{theorem}\n\nWe relegate the proof of Theorem \\ref{theorem: delay} to Appendix \\ref{proof: delay}.\n\n\\textbf{Remark:}\nTheorem \\ref{theorem: delay} implies that predictive offloading conduces to a shorter workload latency; \nin other words, with predicted information, PORA can break the barrier of $[O(1\/V),O(V)]$ power-latency tradeoff.\nFurthermore, the latency reduction induced by PORA is proportional to the inverse of the prediction window size, and approaches zero as prediction window sizes go to infinity. \nIn our simulations, we see that PORA can effectively shorten the average arrival queue latency with only mild-value of future information.\n\n\n\\subsection{Impact of Network Topology}\n\nFog computing systems generally proceed in wireless environments, thus the network topology of such systems is usually dynamic and may change over time slots.\nHowever, at the beginning of each time slot, the network topology is observed and deemed fixed by the end of the time slot. \nTherefore, in the following, we put the focus of our discussion on the impact of network topology within each time slot.\n\nRecall that in our settings, each EFN has access to only a subset of CFNs in its vicinity. For each EFN $i$, the subset of its accessible EFNs is denoted by $\\mathcal{M}_{i}$ with a size of $|\\mathcal{M}_{i}|$. From the perspective of graph theory, we can view the interconnection among fog nodes of different tiers as a directed graph, in which each vertex corresponds to a fog node and each edge indicates a directed connection between nodes. Hence, the value of $|\\mathcal{M}_{i}|$ can be regarded as the out-degree of EFN $i$, which is an important parameter of network topology that measures the number of directed connections originating from EFN $i$. \nDue to time-varying wireless dynamics, the out-degree of each fog node may vary over time slots; \nconsequentially, the resulting topology would significantly affect the system performance. In the following, we discuss such impacts under two channel conditions, respectively.\n\nOn the one hand, within each time slot, poor channel conditions (\\textit{e.g.} in terms of low SINR) would often lead to unreliable or even unavailable connections among fog nodes and hence a network topology with a relatively smaller out-degree of nodes. In this case, each fog node may have a very limited freedom to choose the best target node to offload its workloads, further leading to backlog imbalance among fog nodes or even overloading in its upper tier with a large cumulative queue backlog size. Besides, poor channel conditions may also require more power consumptions to ensure reliable communication between successive fog nodes.\n\nOn the other hand, within each time slot, good channel conditions allow each fog node to have a broader access to the fog nodes in its upper tier, resulting a network topology with a relatively larger out-degree of nodes. In this case, each fog node is able to conduct better decision-making with more freedom in choosing the fog nodes in its upper fog tier, thereby achieving a better tradeoff between power consumptions and backlog sizes.\n\n\\begin{table}[!t]\n\\centering\n\\begin{threeparttable}\n\\caption{Simulation Settings}\n\\label{table: simulation parameters}\n\\begin{tabular}{|c|c|}\n\\hline\nParameter & Value \\\\\n\\hline\n$B$ & $2$ MHz \\\\\n\\hline\n$H_{i,j}(t),\\forall i\\in\\mathcal{N},j\\in\\mathcal{M}$ & $24\\log_{10}d_{i,j}+20\\log_{10}5.8$+60 \\tnote{a} \\\\\n\\hline\n$N_{0}$ & $-174$ dBm\/Hz \\\\\n\\hline\n$P_{i,\\text{max}},\\forall i\\in\\mathcal{N}$ & $500$ mW \\\\\n\\hline\n$L^{(e)}_{i}\\forall i\\in\\mathcal{N}$, $L^{(c)}_{j}\\forall j\\in\\mathcal{M}$ & $297.62$ cycles\/bit \\\\\n\\hline\n$f^{(e)}_{i,\\text{max}},\\forall i\\in\\mathcal{N}$ & $4$ G cycles\/s \\\\\n\\hline\n$f^{(c)}_{j,\\text{max}},\\forall j\\in\\mathcal{M}$ & $8$ G cycles\/s \\\\\n\\hline\n$\\varsigma$ & $10^{-27}$ W$\\cdot$s$^{3}$\/cycle$^{3}$ \\\\\n\\hline\n$b^{(e,l)}_{i,\\text{max}},b^{(e,o)}_{i,\\text{max}},\\forall i\\in\\mathcal{N}$ & $6$ Mb\/s \\\\\n\\hline\n$b^{(c,l)}_{j,\\text{max}},b^{(c,o)}_{j,\\text{max}},\\forall j\\in\\mathcal{M}$ & $12$ Mb\/s \\\\\n\\hline\n$D_{j}(t),\\forall j\\in\\mathcal{M},t$ & $6$ Mb\/s \\\\\n\\hline\n\\end{tabular}\n\\begin{tablenotes}\\footnotesize\n\\item [a] $d_{i,j}$ is the distance between EFN $i$ and CFN $j$.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\n\\subsection{Use Cases}\nIn practice, PORA can be applied as a theoretical framework to design the offloading schemes for fog computing systems under various use cases, such as public safety systems, intelligent transportation, and smart healthcare systems.\nFor example, in a public safety system, each street is usually deployed with multiple smart cameras (IoT devices). \nAt runtime, such smart cameras would upload real-time vision data to one of their accessible EFNs. Each EFN aggregates such data to extract or even analyze the instant road conditions within multiple streets. \nSuch EFNs can upload some of the workload to their upper-layered CFNs (each taking charge of one community consisting of several streets) with greater computing capacities. \nEach CFN can further offload the workload to the cloud via optical fiber links. \nFor latency-sensitive applications, the real-time vision data will be processed locally on EFNs or offloaded to CFNs. \nFor latency-insensitive applications with intensive computation demand, the data will be offloaded to the cloud through the fog nodes. \nPORA conduces to the design of dynamic and online offloading and resource allocation schemes to support such fog systems with various applications.\n\n\n\\section{Numerical Results}\\label{sec: simulation}\n\n\nWe conduct extensive simulations to evaluate PORA and its variants.\nThe parameter settings in our simulation are based on the commonly adopted wireless environment settings that have been used in \\cite{liu2017latency, du2017computation}.\nThe simulation is conducted on a MacBook Pro with 2.3 GHz Intel Core i5 processor and 8GB 2133 MHz LPDDR3 memory, and the simulation program is implemented using Python 3.7.\nThis section firstly presents the basic settings of our simulations, and then provides the key results under perfect and imperfect prediction, respectively.\n\n\n\n\\subsection{Basic Settings}\n\nWe simulate a hierarchal fog computing system with $80$ EFNs and $20$ CFNs. All EFNs have a uniform prediction window size $W$, which varies from $0$ to $30$. Note that $W=0$ refers to the case without prediction. \nFor each EFN $i$, its accessible CFN set $\\mathcal{M}_{i}$ is chosen uniformly randomly from the power set of the CFN set with size $|\\mathcal{M}_{i}|=5$. \nWe set the time slot length $\\tau_{0}=1$ second. \nDuring each time slot, workload arrives to the system in the unit of packets, each with a fixed size of $4096$ bits. \nThe packet arrivals are drawn from previous measurements \\cite{benson2010network}, \nwhere the average flow arrival rate is $538$ flows\/s, \nand the distribution of flow size has a mean of $13$ Kb. \nGiven these settings, the average arrival rate is about $7$ Mbps. \nAll results are averaged over $50000$ time slots. \nWe list all other parameter settings in TABLE \\ref{table: simulation parameters}.\n\n\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{figures1\/decisions_V}\n\t\\caption{Offloading decisions when $W=10$.}\n\t\\label{figure: offloading decisions vs. V}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n \\centering\n \\subfigure[Queue backlogs.]{\n \\label{subfig: backlog-W}\n \\includegraphics[width=0.8\\linewidth]{figures1\/backlog_W}}\n \\subfigure[Power consumptions.]{\n \\label{subfig: power-W}\n \\includegraphics[width=0.8\\linewidth]{figures1\/power_W}}\n \\caption{Performance of PORA vs. $W$ when $V=10^{11}$.}\n \\label{figure: performance vs. window size}\n\\end{figure}\n\n\\subsection{Evaluation with Perfect Prediction}\n\nUnder the perfect prediction settings, we evaluate how the values of parameter $V$ and prediction window size $W$ influence the performance of PORA, respectively.\n\n\n\\textbf{System Performance under Different Values of $V$:}\nFigure \\ref{figure: offloading decisions vs. V} shows the impact of parameter $V$ on the offloading decisions of PORA: \nWhen the value of $V$ is around $10^{10}$, the time-average amount of locally processed workload on EFNs reaches the bottom of the curve, \nwhile other offloading decisions induce the peak workload. \nThe reason is that the offloading decisions are not only determined by the value of $V$, \nbut also influenced by the queue backlog sizes.\n\nFigure \\ref{figure: performance vs. V} presents the impact of the value of $V$ on different types of queues and power consumptions in the system, respectively. \nAs the value of $V$ increases, we see a rising trend in the sizes of all types of queue backlogs, and a roughly falling trend in all types of power consumptions. \n\n\n\n\n\\begin{figure}[!t]\n \\centering\n \\subfigure[Queue backlogs.]{\n \\label{subfig: backlog vs. V}\n \\includegraphics[width=0.8\\linewidth]{figures1\/backlog_V}}\n \\subfigure[Power consumptions.]{\n \\label{subfig: power vs. V}\n \\includegraphics[width=0.8\\linewidth]{figures1\/power_V}}\n \\caption{Performance of PORA when $W=10$.}\n \\label{figure: performance vs. V}\n\\end{figure}\n\n\n\\begin{figure*}[!t]\n \\centering\n \\subfigure[Total queue backlogs.]{\n \\label{subfig: backlog-V-variant}\n \\includegraphics[width=0.4\\linewidth]{figures1\/backlog_V_variant}}\n \\subfigure[Total power consumptions.]{\n \\label{subfig: power-V-variant}\n \\includegraphics[width=0.4\\linewidth]{figures1\/power_V_variant}}\n \\caption{Performance of variants of PORA.}\n \\label{figure: performance v.s. variants}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n \\centering\n \\subfigure[Total queue backlogs.]{\n \\label{subfig: backlog v.s. time}\n \\includegraphics[width=0.4\\linewidth]{figures1\/backlog_time}}\n \\subfigure[Total power consumptions.]{\n \\label{subfig: power v.s. times}\n \\includegraphics[width=0.4\\linewidth]{figures1\/power_time}}\n \\caption{Comparison between PORA and baselines.}\n \\label{figure: comparison}\n\\end{figure*}\n\n\n\\textbf{System Performance with Different Values of Prediction Window Size $W$:}\nFigures \\ref{subfig: backlog-W} and \\ref{subfig: power-W} show the system performance with the prediction window size $W$ varying from $0$ to $30$. \nWith perfect prediction, PORA effectively shortens the average queueing latencies on EFN arrival queues -- eventually close to zero with no extra power consumption and only a mild-value of prediction window size ($W=20$ in this case).\n\n\n\\textbf{PORA vs. PORA-$d$ (Low-Sampling Variant):}\nIn practice, since PORA requires to sample system dynamics across various fog nodes, it may incur considerable sampling overheads.\nBy adopting the idea of randomized load balancing techniques \\cite{mitzenmacher2001power}, we propose PORA-$d$, a variant of PORA that reduces the sampling overheads by \nprobing $d$ ($d\\in\\{1,2,3,4\\}$) \\footnote{When $d=1$, the scheme degenerates to uniform random sampling.} CFNs and conducting resource allocation on which are uniformly chosen for each EFN from its accessible CFN set. \n\nFigure \\ref{figure: performance v.s. variants} compares the performance of PORA with PORA-$d$. \nWe observe that PORA achieves the smallest queue backlog size. The result is reasonable since each EFN has access to 5 CFNs under PORA, more than the $d\\leq 4$ CFNs under PORA-$d$. As a result, each EFN has more chance to access to the CFNs with better wireless channel condition and processing capacity under PORA when compared with PORA-$d$. The observation that the queue backlog size increases as $d$ decreases further verifies our analysis. In fact, we can view $d$ as the degree of each EFN in the network topology. As $d$ decreases, the system performance degrades.\nHowever, when the value of $V$ is sufficiently large, PORA-$d$ achieves the similar power consumptions as PORA and the ratio of increment in the backlog size is small. For example, when $V=2\\times 10^{11}$, PORA-$4$ achieves $4.3$\\% larger backlog size than PORA, and PORA-$3$ achieves $10.9$\\% larger backlog size than PORA.\nIn summary, PORA-$d$ (when $d=2,3,4$) can reduce the sampling overheads by trading off only a little performance degradation under large $V$.\n\n\n\\textbf{Comparison of PORA and Baselines:}\nWe introduce four baselines to evaluate the performance of PORA: (1) NOL (No Offloading): All nodes in the EFT process packets locally. (2) O2CFT (Offload to CFT): All packets are offloaded to the CFT and processed therein. (3) O2CLOUD (Offload to Cloud): All packets are offloaded to the cloud. (4) RANDOM: Each fog node randomly chooses to offload each packet or process it locally with equal chance. \nNote that all above baselines are also assumed capable of pre-serving future workloads in the prediction window.\nFigure \\ref{figure: comparison} compares the instant total queue backlog sizes and power consumptions over time slots under the five schemes (PORA, NOL, O2CFT, O2CLOUD, RANDOM), \nwhere $W=10$ and $V\\in\\{10^{9},10^{11}\\}$.\n\n\nWe observe that scheme O2CLOUD achieves the minimum power consumptions, but incurs constantly increasing queue backlog sizes over time. \nThe reasons are shown as follows. \nOn one hand, in our settings, the mean power consumption for transmitting workload from EFT to CFT is smaller than the mean power consumption of processing the same amount of workload on fog nodes; under scheme O2CLOUD, only wireless transmit power is consumed and hence the minimum is achieved. \nOn the other hand, all the workload must travel through all fog tiers before being offloaded to the cloud, which results in network congestion within fog tiers and thus workload accumulation with increasing queue backlogs.\n\nAs Figure \\ref{figure: comparison} illustrates, PORA achieves the maximum power consumptions but the smallest backlog size when $V=10^{9}$. \nUpon convergence of PORA, the power consumptions under all these schemes reach the same level, but the differences between their queue backlog sizes become more obvious:\nPORA ($V=10^{9}$) reduces $96$\\% of the queue backlog when compared with NOL and RANDOM.\nThe results demonstrate that with the appropriate choice of the value of $V$,\nPORA can achieve less latency than the four baselines under the same power consumptions.\n\n\\subsection{Evaluation with Imperfect Prediction}\n\nIn practice, prediction errors are inevitable. \nHence, we investigate the performance of PORA in the presence of prediction errors \\cite{chen2017timely}. \nParticularly, we consider two kinds of prediction errors: false alarm and missed detection.\nA packet is falsely alarmed if it is predicted to arrive but it does not arrive actually.\nA packet is missed to be detected if it will arrive but is not predicted.\nWe assume that all EFNs have the uniform false-alarm rate $p_{1}$ and missed-detection rate $p_{2}$.\nIn our simulation, we consider different pairs of values of $(p_{1},p_{2})$: $(0.0, 0.0)$, $(0.05, 0.05)$, $(0.5, 0.05)$, $(0.05, 0.25)$, and $(0.5, 0.25)$. \nNote that $(p_{1},p_{2})=(0.0,0.0)$ corresponds to the case when the prediction is perfect. \n\n\\begin{figure}[!t]\n \\centering\n \\subfigure[Total queue backlogs.]{\n \\label{subfig: backlog under imperfect prediction}\n \\includegraphics[width=0.8\\linewidth]{figures1\/backlog_V_error}}\n \\subfigure[Total power consumptions.]{\n \\label{subfig: power under imperfect prediction}\n \\includegraphics[width=0.8\\linewidth]{figures1\/power_V_error}}\n \\caption{Performance of PORA under imperfect prediction.}\n \\label{figure: performance under imperfect prediction}\n\\end{figure}\n\nFigure \\ref{figure: performance under imperfect prediction} presents the results under prediction window size $W=10$.\nWe observe when $V\\leq 7.5\\times 10^{10}$, both the total queue backlog sizes and power consumptions under imperfect prediction are larger than that under perfect prediction. The reason for this performance degradation is twofold:\nFirst, arrivals that are missed to be detected cannot be pre-served, thus leading to larger queue backlog sizes.\nSecond, PORA allocates redundant resources to handle the falsely predicted arrivals, thus causing more power consumptions.\nAs the value of $V$ increases, this performance degradation becomes negligible. Taking the total queue backlog under $(p_{1},p_{2})=(0.25,0.5)$ as an example, when compared with the case under perfect prediction, it increases by $4.72\\%$ at $V=10^{11}$, and increases by $2.24\\%$ at $V=2\\times 10^{11}$. \nMoreover, there is no extra power consumption under imperfect prediction when $V\\geq 7.5\\times 10^{10}$ since PORA tends to reserve resources to reduce power consumptions under large $V$.\n\nIn summary, there will be performance degradation in both total queue backlog sizes and power consumptions in the presence of prediction errors. However, as the value of $V$ increases, this degradation decreases and becomes negligible.\nThough a large value of $V$ can improve the robustness of PORA and achieve small power consumptions, it brings long workload latencies. \nIn practice, the choice of the value of $V$ depends on how the system designer trades off all these criterions.\n\n\\section{Conclusion}\\label{sec: conclusion}\n\nIn this paper, we studied the problem of dynamic offloading and resource allocation with prediction in a fog computing system with multiple tiers. \nBy formulating it as a stochastic network optimization problem, we proposed PORA, an efficient online scheme that exploits predictive offloading to minimize power consumption with queue stability guarantee. \nOur theoretical analysis and trace-driven simulations showed that PORA achieves a tunable power-latency tradeoff, while effectively shortening latency with only mild-value of future information, even in the presence of prediction errors. \nAs for future work, our model can be further extended to more general settings such that the instant wireless channel states may be unknown by the moment of decision making or the underlying system dynamics is non-stationary.\n\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n \\newpage\n\\fi\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\nThe recently proposed model of quantum gravity by Ho\\v{r}ava \\cite{horava1,horava2,horava3} has recently attracted much attention, and many aspects of it\n have been extensively analyzed, ranging from formal developments, cosmology, dark energy\n and dark matter, spherically symmetric solutions, gravitational waves, and its viability with observational constraints; for a full list of references see, e.g., \\cite{cinesi,Bom,Wang,Carloni}.\nSuch a theory admits Lifshitz's scale invariance: $\\mathbf{x} \\rightarrow b \\mathbf{x}, \\quad t \\rightarrow b^{q} t$, and, after this, it is referred to as Ho\\v{r}ava-Lifshitz (HL) gravity. Actually, it has anistropic scaling in the short distances domain (UV), since it is $q=3$, while isotropy is recovered at large distances (IR).\n\nOne of the key features of the theory is its good UV behavior, since it is power-counting renormalizable; for a discussion of the renormalizability beyond power counting arguments, see \\cite{Orla}. However, in its original formulation, it experiences some problems: for instance, it leads to a non-zero cosmological constant with the wrong sign, in order to be in agreement with the observations \\cite{Soda,horatius,visser1}. To circumvent these issues, it was suggested to abandon the principle of ``detailed balance'' \\cite{Calca1,Calca2}, initially introduced by Ho\\v{r}ava in his model to restrict the number of possible parameters. As a consequence, phenomenologically viable extensions of the theory were proposed \\cite{visser1,visser2}. It was also shown that HL gravity can reproduce General Relativity (GR) at large distances \\cite{pope,KS}; for other solutions non-asymptotically flat see \\cite{Cai1,Cai2}. However, there is still an ongoing discussion on the consistency of HL gravity, since it seems that modes arise which develop exponential instabilities at short distances, or become strongly coupled \\cite{padilla,blas}. Moreover, according to \\cite{Miao}, the constraint algebra does not form a closed structure. Perturbative instabilities affecting HL gravity have been pointed out in \\cite{Saridakis}.\n\nActually, it is important to stress that, up to now, in HL gravity the gravitational field is purely geometrical: in other words, the way matter has to be embedded still needs to be studied. Nevertheless, there are interesting vacuum solutions that can be studied, such as the static spherically symmetric solution found by Kehagias and Sfetsos (hereafter KS) \\cite{KS}. Such a solution is the analog of Schwarzschild solution of GR and, moreover, it asymptotically reproduces the usual behavior of Schwarzschild spacetime. It is interesting to point out that it is obtained without requiring the projectability condition, assumed in the original HL theory, while spherically symmetric solutions with the projectability condition are however available \\cite{tang,Wang}. {Nonetheless, because of its simplicity, it is possible to consider KS solution as toy model useful to better understand some phenomenological implications of HL gravity. Actually, in \\cite{Tib} it was shown that KS solution is in agreement with the classical tests of GR, while in a previous paper \\cite{HLIR} we studied the corrections to the general relativistic Einstein's pericentre precession determined by this solution and compared the theoretical predictions to\nthe latest determinations of the corrections to the standard Newtonian\/Einsteinian planetary perihelion precessions recently estimated with the EPM2008 ephemerides. We found that the KS dimensionless parameter is constrained from the bottom at $\\omega_0\\geq 10^{-12}-10^{-24}$ level depending on the planet considered.\n\nIn our analysis, we assumed that particles followed geodesics of KS metric: however, it is important to point out that this is true if matter is minimally and universally coupled to the metric, which is not necessarily true in HL gravity, where, as we said above, the role of matter has not been yet clarified. In this paper, starting from the same assumption, we focus on the effects induced by the examined solution on the orbital period $P_{\\rm b}$ of a test particle, on an extra solar system environment. We will explicitly work out the consequent correction $P_{\\omega_0}$ to the usual third Kepler law in Section \\ref{3kep}. In Section \\ref{osiris} we compare it with the observations of the transiting extrasolar planet HD209458b \\virg{Osiris}. {We point out that the resulting constraints are to be considered as preliminary and just order-of-magnitude figures because, actually, the entire data set of HD209458b should be re-processed again by explicitly modeling the effect of the KS gravity; however, this is outside the scopes of the present paper.} Section \\ref{conclu} is devoted to the conclusions.\n\\section{KS corrections to the third Kepler law}\\lb{3kep}\nAs shown in \\cite{HLIR},\nfrom \\cite{Brum}\n\\begin{eqnarray} \\ddot x^i &=& -\\frac{1}{2}c^2 h_{00,i} - \\frac{1}{2}c^2 h_{ik}h_{00,k}+h_{00,k}\\dot x^k\\dot x^i \\nonumber \\\\ &+& \\left(h_{ik,m}-\\frac{1}{2}h_{km,i}\\right)\\dot x^k\\dot x^m,\\ i=1,2,3,\\end{eqnarray}\nit is possible to obtain the following radial acceleration acting upon a test particle at distance $d$ from a central body of mass $M$\n\\eqi \\vec{A}_{\\omega_0}\\approx \\frac{4 (GM)^4}{\\omega_0 c^6 d^5}\\hat{d},\\lb{accel}\\eqf valid up to terms of order $\\mathcal{O}(v^2\/c^2)$.\nIts effect on the pericentre of a test particle have been worked out in \\cite{HLIR}; here we want to look at a different orbital feature affected by \\rfr{accel} which can be compared to certain observational determinations.\n\nThe mean anomaly is defined as\n\\eqi \\mathcal{M}\\doteq n(t-t_p);\\eqf in it $n=\\sqrt{GM\/a^3}$ is the Keplerian mean motion, $a$ is the semimajor axis and $t_p$ is the time of pericentre passage.\nThe anomalistic period $P_{\\rm b}$ is the time elapsed between two consecutive pericentre passages; for an unperturbed Keplerian orbit it is $P_{\\rm b}=2\\pi\/n$. Its modification due to a small perturbation of the Newtonian monopole can be evaluated with standard perturbative approaches. The Gauss equation for the variation of the mean anomaly is, in the case of a radial perturbation $A_d$ to the Newtonian monopole \\cite{Ber},\n\\eqi \\frac{d{\\mathcal{M}}}{dt}=n-\\rp{2}{na}A_d\\left(\\rp{d}{a}\\right)+\\rp{(1-e^2)}{nae}A_d\\cos f,\\lb{Gaus}\\eqf where $e$ is the eccentricity and $f$ is the true anomaly counted from the pericentre position.\nThe right-hand-side of \\rfr{Gaus} has to be valuated onto the unperturbed Keplerian orbit given by (see \\cite{Roy})\n\\eqi d=\\rp{a(1-e^2)}{1+e\\cos f}.\\eqf\nBy using (see \\cite{Roy})\n\\eqi df = \\left(\\rp{a}{d}\\right)^2(1-e^2)^{1\/2}d\\mathcal{M}\\eqf\nand\n\\eqi \\int_0^{2\\pi} (1+e\\cos f)^2\\left[2-\\rp{(1+e\\cos f)}{e}\\cos f\\right]df=\\pi\\left(1+\\rp{5}{4}e^2\\right),\\eqf it is possible to work out the correction to the Keplerian period due to \\rfr{accel}; it is\n\\eqi P_{\\omega_0}=\\rp{4\\pi (GM)^4\\left(1+\\rp{5}{4}e^2\\right)}{\\omega_0 c^6 n^3 a^6(1-e^2)^{5\/2}}=\\rp{4\\pi (GM)^{5\/2}\\left(1+\\rp{5}{4}e^2\\right)}{\\omega_0 c^6 a^{3\/2}(1-e^2)^{5\/2}}.\\lb{PHL}\\eqf\nNote that \\rfr{PHL} retains its validity in the limit $e\\rightarrow 0$ becoming equal to\n\\eqi P_{\\omega_0}\\rightarrow\\rp{4\\pi (GM)^{5\/2}}{\\omega_0 c^6 d^{3\/2}}\\lb{circo},\\eqf where $d$ represents now the fixed radius of the circular orbit. It turns out that \\rfr{circo} is equal to the expression that can be easily obtained by equating the centripetal acceleration $\\Omega^2 d$, where $\\Omega$ is the particle's angular speed, to the total gravitational acceleration $GM\/d^2 - 4(GM)^4\/\\omega_0c^6 d^5$ with the obvious assumption that the Newtonian monopole is the dominant term in the sum.\n\\section{Confrontation with the observations}\\lb{osiris}\nIn the scientific literature there is a large number of papers (see, e.g., \\cite{Tal,Wri,Ove,Jae,Rey,Ser1,Ser2,Ior,Bro,Peri,Mof,Adl,Page}) in which the authors use the third Kepler law to determine, or, at least, constrain un-modeled dynamical effects of mundane, i.e. due to the standard Newtonian\/Einsteinian laws of gravitation, or non-standard, i.e. induced by putative modified models of gravity.\nAs explained below, in many cases such a strategy has been, perhaps, followed in a self-contradictory way, so that the resulting constraints on, e.g., new physics, may be regarded as somewhat \\virg{tautologic}.\n\nLet us briefly recall that the orbital period $P_{\\rm b}$ of two point-like bodies of mass $m_1$ and $m_2$ is, according to the third Kepler law,\n\\eqi P^{\\rm Kep}=2\\pi\\sqrt{\\rp{a^3}{G{M}}},\\eqf\nwhere $a$ is the relative semi-major axis and ${M}=m_1+m_2$ is the total mass of the system.\nLet us consider an unmodeled dynamical effect which induces a non-Keplerian (NK) correction to the third Kepler law, i.e.\n\\eqi P_{\\rm b}= P^{\\rm Kep} + P^{\\rm NK},\\eqf\nwhere\n\\eqi P^{\\rm NK}=P^{\\rm NK}(M,a,e; p_j),\\eqf\nis the analytic expression of the correction to the third Kepler law in which $p_j$, $j=1,2,...$N are the parameters of the NK effect to be determined or constrained. Concerning standard physics, $P_{\\rm NK}$ may be due to the centrifugal oblateness of the primary, tidal distortions, General Relativity; however, the most interesting case is that in which $P_{\\rm NK}$ is due to some putative modified models of gravity. {As a first, relatively simple step to gain insights into the NK effect one can act as follows.}\nBy comparing the measured orbital period to the computed Keplerian one it is possible, in principle, to obtain preliminary information on the dynamical effect investigated from $\\Delta P\\doteq P_{\\rm b}-P^{\\rm Kep}$. {Actually, one should re-process the entire data set of the system considered by explicitly modeling the non-standard gravity forces, and simultaneously solving for one or more dedicated parameter(s) in a new global solution along with the other ones routinely estimated. Such a procedure would be, in general, very time-consuming and should be repeated for each models considered. Anyway, it is outside the scopes of the present paper, but it could be pursued in further investigations.}\n\nConcerning our simple approach, in order to meaningfully solve for $p_j$ in\n\\eqi\\Delta P=P^{\\rm NK}\\eqf\nit is necessary that\n\\begin{itemize}\n \\item In the system considered a measurable quantity which can be identified with the orbital period and directly measured independently of the third Kepler law itself, for example from spectroscopic or photometric measurements, must exist. This is no so obvious as it might seem at first sight; indeed, in a N-body system like, e.g., our solar system a directly measurable thing like an \\virg{orbital period} simply does not exist because the orbits of the planets are not closed due to the non-negligible mutual perturbations. Instead, many authors use values of the \\virg{orbital periods} of the planets which are retrieved just from the third Kepler law itself.\n Examples of systems in which there is a measured orbital period are many transiting exoplanets, binaries and, e.g, the double pulsar.\n Moreover, if the system considered follows an eccentric path one should be careful in identifying the measured orbital period with the predicted sidereal or anomalistic periods. A work whose authors are aware of such issues is \\cite{Capoz}.\n \n \n \\item The quantities entering $P^{\\rm Kep}$, i.e. the relative semimajor axis $a$ and the total mass $M$, must be known independently of the third Kepler law. Instead, in many cases values of the masses obtained by applying just the third Kepler law itself are used.\n Thus, for many exoplanetary systems the mass $m_1\\doteq M_\\bigstar$ of the hosting star should be taken from stellar evolution models and the associated scatter should be used to evaluate the uncertainty $\\delta M_\\bigstar$ in it, while for the mass $m_2=m_p$ of the planet a reasonable range of values should be used instead of straightforwardly taking the published value because it comes from the mass function which is just another form of the third Kepler law.\n {Some extrasolar planetary systems represent good scenarios because it is possible to know many of the parameters entering $P^{\\rm Kep}$ independently of the third Kepler law itself, thanks to the redundancy offered by the various techniques used.}\n \n\nSuch issues have been accounted for in several astronomical and astrophysical scenarios in, e.g., \\cite{IorWD,IorIJMPA,IorCyg,IorReg,IorNA}.\n\\end{itemize}\n\n\\section{The transiting exoplanet HD209458b}\nLet us consider HD 209458b \\virg{Osiris}, which is the first exoplanet\\footnote{See on the WEB http:\/\/www.exoplanet.eu\/} discovered with the transit method \\cite{Cha,Hen}. Its orbital period $P_{\\rm b}$ is known with a so high level of accuracy that it was proposed to use it for the first time to test General Relativity in a planetary system different from ours \\cite{osi}; for other proposals to test General Relativity with different orbital parameters of other exoplanets, see \\cite{ada,ung1,ung2,RAGO}.\n\nIn the present case, the system's parameters entering the Keplerian period\ni.e. the relative semimajor axis $a$, the mass $M_{\\bigstar}$ of the host star and the mass $m_p$ of the planet, can be determined independently of the third Kepler law itself, so that it is meaningful to compare the photometrically measured orbital period $P_{\\rm b}=3.524746$ d \\cite{exo} to the computed Keplerian one $P^{\\rm Kep}$: their difference can be used to put genuine constraints on KS solution which predicts the corrections of \\rfr{PHL} to the third Kepler law.\nIndeed, the mass $M_{\\bigstar}=1.119\\pm 0.033$M$_{\\odot}$ and the radius $R_{\\bigstar}=1.155^{+0.014}_{-0.016}$R$_{\\odot}$ of the star \\cite{exo}, along with other stellar properties, are fairly straightforwardly estimated by matching direct spectral observations with stellar evolution models since for HD 209458 we have also the Hipparcos parallax $\\pi_{\\rm Hip}=21.24\\pm 1.00$ mas \\cite{Perry}.\nThe semimajor axis-to-stellar radius ratio $a\/R_{\\bigstar}=8.76\\pm 0.04$ is estimated from the photometric light curve, so that $a=0.04707^{+0.00046}_{-0.00047}$AU \\cite{exo}. The mass $m_p$ of the planet can be retrieved from the parameters of the photometric light curve and of the spectroscopic one entering the formula for the planet's surface gravity $g_p$ (eq.(6) in \\cite{exo}). As a result, after having computed the uncertainty in the Keplerian period by summing in quadrature the errors due to $\\delta a,\\delta M_{\\bigstar},\\delta m_p$, it turns out\n \\eqi \\Delta P\\doteq P_{\\rm b}-P^{\\rm Kep} = 204\\pm 5354\\ {\\rm s};\\lb{exop}\\eqf the uncertainties $\\delta M_{\\bigstar}$, $\\delta a$, $\\delta m_p$ contribute 4484.88 s, 2924.77 s, 2.66 s, respectively to $\\delta(\\Delta P)=5354$ s.\n\n The discrepancy $\\Delta P$ between $P_{\\rm b}$ and $P^{\\rm Kep}$ of \\rfr{exop} is statistically compatible with zero; thus, \\rfr{exop} allows to constrain the parameter $\\omega_0$ entering $P_{\\omega_0}$.\n Since\n \\eqi P^{\\rm NK}\\doteq P_{\\omega_0}=\\rp{\\mathcal{K}}{\\omega_0},\\eqf\n with\n \\eqi \\mathcal{K}\\doteq\\rp{4\\pi(GM)^{5\/2}}{c^6 d^{3\/2}}=8\\times 10^{-15}\\ {\\rm s},\\lb{cazza}\\eqf\n by equating the non-Keplerian correction $P_{\\omega_0}$ to the measured $\\Delta P$ one has\n \\eqi \\omega_0=\\rp{\\mathcal{K}}{\\Delta P}.\\lb{vaffa}\\eqf\n Since $\\Delta P$ is statistically compatible with zero, the largest value of $\\omega_0$ is infinity; from \\rfr{vaffa} a lower bound on $|\\omega_0|$ can be obtained amounting to\\eqi |\\omega_0|\\geq 1.4\\times 10^{-18}.\\lb{lowerbo}\\eqf\n A confrontation with the solar system constraints\\footnote{To avoid confusions with the perihelion $\\omega$, the KS parameter is dubbed $\\psi_0$ in \\cite{HLIR}.} Our previous paper \\cite{HLIR} shows that such a lower bound is at the level of those from Jupiter and Saturn, while it contradicts the possibility that values of $\\omega_0$ as small as those allowed by Uranus, Neptune and Pluto ($|\\omega_0|\\geq 10^{-24}-10^{-22}$) may exist.\n However, tighter constraints are established by the inner planets for which $|\\omega_0|\\geq 10^{-15}-10^{-12}$.\n\n\n\n\n\\section{Conclusions}\\lb{conclu}\nWe have investigated how the third Kepler law is modified by the KS solution,\nwhose Newtonian and lowest order post-Newtonian limits coincides with those of GR, by using the standard Gauss perturbative approach. The resulting expression for $P_{\\omega_0}$, obtained from the Gauss equation of the variation of the mean anomaly $\\mathcal{M}$, in the limit $e\\rightarrow 0$ reduces to the simple formula which can be derived by equating the centripetal acceleration to the Newton$+$KS gravitational acceleration for a circular orbit.\n\nThen, after having discussed certain subtleties connected, in general, with a meaningful use of the third Kepler law to put on the test alternative theories of gravity, we compared our explicit expression for $P_{\\omega_0}$ to the discrepancy $\\Delta P$ between the phenomenologically determined orbital periods $P_{\\rm b}$ and the computed Keplerian ones $P^{\\rm Kep}$ for the transiting extrasolar planet HD209458b \\virg{Osiris}. Since $\\Delta P$ is statistically compatible with zero, it has been possible to {preliminary} obtain the lower bound $|\\omega_0|\\geq 1.4 \\times 10^{-18}$ on the dimensionless KS parameter. {However, the entire data set of HD209458b should be re-processed by including KS gravity as well, and a dedicated, solve-for parameter should be estimated as well. The previously reported } constraint rules out certain smaller values allowed by the lower bounds obtained from the perihelia of Uranus, Neptune and Pluto ($|\\omega_0|\\geq 10^{-24}-10^{-22}$). On the other hand, our exoplanet bound still leaves room for values of $\\omega_0$ too small according to the constraints from the perihelia of Mercury, Venus and the Earth ($|\\omega_0|\\geq 10^{-14}-10^{-12}$).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\paragraph{Planar maps.}\n\nMaps, i.e. gluings of polygons forming an orientable surface, have been the object of extensive research in the last decades, both from the combinatorial and probabilistic viewpoints. The most popular category of maps are planar maps, i.e. maps homeomorphic to the sphere. Their combinatorial study goes back to Tutte in the 60s, e.g.~\\cite{Tutte63}, who gave explicit formulas for the enumeration of various classes of planar maps using a generating functions approach. More recently, bijective approaches have been developped such as the Cori--Vauquelin--Schaeffer bijection for quadrangulations~\\cite{Sch98} and its generalization, the Bouttier--di Francesco--Guitter bijection~\\cite{BDG04}.\n \nOn the other hand, much attention has been given in the last 20 years to asymptotic properties of large random planar maps picked uniformly in certain classes. These asymptotic properties are usually understood by proving the convergence of random maps in some sense when the size goes to infinity. Two different notions of limits are commonly used: scaling and local limits. Scaling limits, which we will not study in this work, consist in renormalizing the distances in order to build continuous objects. In particular, many discrete models are known to have the Brownian map as a scaling limit~\\cite{LG11, Mie11, Mar16}. The theory of scaling limits of planar maps shares deep links with other random geometry models such as Liouville Quantum Gravity~\\cite{MS16b}. On the other hand, local limits, on which the present work focuses, study the neighbourhood of a typical point in a map in order to obtain an infinite but discrete object in the limit. In the context of planar maps, this was first considered by Angel and Schramm who proved the convergence of large uniform triangulations towards the Uniform Infinite Planar Triangulation (UIPT)~\\cite{AS03}. The study of the UIPT using Markovian explorations called \\emph{peeling} explorations was then initiated by Angel~\\cite{Ang03}. More general models have followed since, such as general planar maps with Boltzmann weights on the face degrees~\\cite{St18}. The bipartite case, which will be of particular interest for us, is investigated in~\\cite{Bud15, BC16}, see also~\\cite{C-StFlour} for a complete survey.\n\n\\paragraph{Maps of higher genus.}\nIt seems natural to try to extend the combinatorial and probabilistic study of planar maps to maps of higher genus. On the combinatorial side, the enumeration of maps with any genus is a very rich topic, with links to irreducible representations of the symetric group and integrable hierarchies~\\cite{MJD00,Okounkov00}. In particular, double recursions on both the size and the genus are known for the counting of maps, see~\\cite{GJ08} for triangulations and \\cite{Lo19} for general classes of bipartite maps. However, explicit enumeration formulas are lacking. Asymptotics can be obtained when the genus is fixed and the size goes to infinity~\\cite{BC86}, but are still missing when the genus goes to infinity as well.\n\nSimilarly, on the probabilistic side, higher genus versions of random surface models have been constructed such as Brownian surfaces~\\cite{Bet16} or Liouville quantum gravity on complex tori \\cite{DRV16}. However, their behaviour when the genus goes to infinity is still poorly understood. Finally, a regime that is much easier to handle is the regime where the genus is not constrained, and the faces are simply glued uniformly at random \\cite{Gam06, CP16, BCP19}. In this case, the genus is concentrated very close to its maximal possible value.\n\nMore recently, some progress was made in the study of high genus maps, namely when the genus grows linearly in the size of the map. In this case, the Euler formula shows that maps satisfy a discrete notion of \"negative average curvature\", which suggests that the neighbourhood of a typical vertex should look hyperbolic. The first category of maps that was investigated in this setting were uniform unicellular maps (i.e. maps with one face). See \\cite{ACCR13} for the proof of local convergence to a supercritical Galton--Watson tree, and \\cite{Ray13a} for the study of more global properties such as logarithmic diameter.\n\nShortly after, Curien introduced a one-parameter family of random hyperbolic triangulations of the plane \\cite{CurPSHIT}, following the work of Angel and Ray in the half-planar case \\cite{AR13}. More precisely, random maps of this family are called Planar Stochastic Hyperbolic Triangulations (PSHT) $\\left( \\mathbb{T}_{\\lambda} \\right)_{0<\\lambda \\leq (12\\sqrt{3})^{-1}}$ and they are the only random triangulations satisfying the following spatial Markov property: for any finite triangulation $t$ with $|t|$ vertices in total and a hole of perimeter $p$, we have\n\\begin{equation}\n\\P \\left( t \\subset \\mathbb{T}_{\\lambda} \\right) = C_{p}(\\lambda) \\lambda^{|t|}.\n\\end{equation}\nIn particular, such a triangulation exists if and only if $\\lambda \\in \\left( 0, \\frac{1}{12\\sqrt{3}} \\right]$. Except for the critical case $\\lambda=\\frac{1}{12\\sqrt{3}}$ (which corresponds to the UIPT), these objects exhibit hyperbolic properties \\cite{CurPSHIT, B18}. Benjamini and Curien conjectured in~\\cite{CurPSHIT} that they are the local limits of uniform high genus triangulations. \n\nIn a recent paper \\cite{BL19}, the authors of the present work proved this conjecture. Asymptotics for the enumeration of high genus triangulations up to subexponential factors were derived as a byproduct.\n\n\\paragraph{Infinite Boltzmann Planar Maps.}\nThe goal of the present work is to generalize the results of~\\cite{BL19} to a much wider family of models, where faces do not have to be triangles. For combinatorial reasons\\footnote{More precisely, the enumeration results in the planar case are simpler for bipartite maps, and the recursion of~\\cite{Lo19} holds only for bipartite maps.}, we will restrict ourselves to bipartite maps, which means that the face degrees have to be even. The limiting objects appearing in the limits are the \\emph{Infinite Boltzmann Bipartite Planar Maps} (IBPM) introduced in~\\cite[Appendix C]{B18these} as an analog of the PSHT for bipartite maps. We also refer to~\\cite[Chapter 8]{C-StFlour} for the study of basic properties of these objects.\n\nThe IBPM are characterized by a spatial Markov property similar to the one satisfied by the PSHT. If $m$ is a finite map with one hole, we write $m \\subset M$ if $M$ can be obtained by filling the hole of $m$, possibly with a map with a nonsimple boundary\\footnote{More precisely, we use the sense introduced by Budd in~\\cite{Bud15}, i.e. the sense corresponding to the \"lazy\" peeling process, as opposed to the one introduced by Angel in~\\cite{Ang03}. See Section~\\ref{subsec_univ_defns} for precise definitions.}. Let $\\mathbf{q}=(q_j)_{j \\geq 1}$ be a sequence of nonnegative numbers. An infinite random planar map $M$ is called a $\\mathbf{q}$-IBPM if there are numbers $\\left( C_p \\right)_{p \\geq 1}$ such that, for any finite map $m$ with one hole of perimeter $2p$, we have\n\\[ \\P \\left( m \\subset M \\right) = C_p \\times \\prod_{f \\in m} q_{\\mathrm{deg}(f)\/2}, \\]\nwhere the product is over all internal faces of $m$. In particular, they generalize the infinite \\emph{critical} bipartite Boltzmann maps defined and studied by Budd in \\cite{Bud15}. It was proved in~\\cite[Appendix C]{B18these} that there is at most one $\\mathbf{q}$-IBPM, that we denote by $\\mathbb{M}_{\\mathbf{q}}$. Moreover \\cite[Appendix C]{B18these} provides both necessary conditions and sufficient ones on $\\mathbf{q}$ for the existence of $\\mathbb{M}_{\\mathbf{q}}$, but no explicit characterization (these results are recalled in Section~\\ref{subsec_IBPM} below). The present work improves on these results by giving an explicit parametrization of the weight families $\\mathbf{q}$ for which $M_{\\mathbf{q}}$ exists. More precisely, as stated in Theorem~\\ref{thm_prametrization_rootface} below, such families $\\mathbf{q}$ can be parametrized by the law of the degree of the root face of $\\mathbb M_{\\mathbf{q}}$, which may be any law on $\\{2,4,6,\\dots\\}$, and an additional hyperbolicity parameter $\\omega \\in [1, +\\infty)$. The critical maps of~\\cite{Bud15} correspond to the case $\\omega=1$, and are already known to be local limits of \\emph{planar} maps. On the other hand, the case $\\omega>1$ has a hyperbolic flavour.\n\n\\paragraph{Local limits of high genus bipartite maps.}\nThe main result of this work is that uniform maps with high genus and prescribed face degrees converge locally to the $\\mathbf{q}$-IBPM when the size goes to infinity. This can be seen as a \\textit{universality} result in the domain of high genus maps, in the sense that regardless of the precise model of maps, the same phenomenon is observed.\n\nMore precisely, we will use the notation $\\mathbf{f}=(f_j)_{j \\geq 1}$ for face degree sequences (i.e. $f_j \\geq 0$ for all $j$, and $f_j=0$ eventually). For such a sequence $\\mathbf{f}$, we set $|\\mathbf{f}|=\\sum_{j \\geq 1} j f_j$, which describes the number of edges of a map with $f_j$ faces of degree $2j$ for all $j \\geq 1$. For $g \\geq 0$, we also write\n\\begin{equation}\\label{defn_v_f_g}\nv(\\mathbf{f}, g) = 2-2g+\\sum_{j \\geq 1} (j-1)f_j.\n\\end{equation}\nBy the Euler formula, a bipartite map with genus $g$ and face degrees described by $\\mathbf{f}$ exists if and only if $v(\\mathbf{f}, g) \\geq 2$, and in this case $v(\\mathbf{f}, g)$ is the number of vertices of such a map. For such $\\mathbf{f}$ and $g$, we denote by $M_{\\mathbf{f},g}$ a uniform bipartite map with genus $g$ and $f_j$ faces of degree $2j$ for all $j \\geq 1$.\n\n\\begin{thm}\\label{univ_main_thm}\nLet $(\\mathbf{f}^{n})_{n \\geq 1}$ be a sequence of face degree sequences, and let $(g_n)$ be a sequence such that $v(\\mathbf{f}^{n}, g_n) \\geq 2$ for all $n \\geq 1$. We assume that $|\\mathbf{f}^n| \\to +\\infty$ when $n \\to +\\infty$ and that $\\frac{f^n_j}{|\\mathbf{f}^n|} \\to \\alpha_j$ for all $j \\geq 1$, where $\\sum_{j \\geq 1} j \\alpha_j=1$.\nWe also assume $\\frac{g_n}{|\\mathbf{f}^n|} \\to \\theta$, where $0 \\leq \\theta < \\frac{1}{2} \\sum_{j \\geq 1} (j-1) \\alpha_j$.\nFinally, assume that $\\sum_{j \\geq 1} j^2 \\alpha_j <+\\infty$.\n\nThen we have the convergence in distribution\n\\[ M_{\\mathbf{f}^n, g_n} \\xrightarrow[n \\to +\\infty]{(d)} \\mathbb M_{\\mathbf{q}}\\]\nfor the local topology, where the weight sequence $\\mathbf{q}$ depends only on $\\theta$ and $\\left( \\alpha_j \\right)_{j \\geq 1}$, in an injective way.\n\\end{thm}\n\nLet us now make a few comments on the various assumptions of the main theorem.\n\\begin{itemize}\n\\item[$\\bullet$]\nWe first note that the assumption that $\\sum_{j \\geq 1} j \\alpha_j=1$ means that the proportion of the edges that are incident to a face with degree larger than $A$ goes to $0$ as $A \\to +\\infty$, uniformly in $n$. This is equivalent to saying that the root face stays almost surely finite in the limit, so this assumption is necessary to obtain a local limit with finite faces. If this assumption is waived, we expect to obtain different limit objects with infinitely many infinite faces.\n\\item[$\\bullet$]\nThe assumption $\\theta < \\frac{1}{2} \\sum_{j \\geq 1} (j-1) \\alpha_j$ means that the number of vertices is roughly proportional to $|\\mathbf{f}^n|$, so that the average degree of the vertices stays bounded. Therefore, it is also necessary in order to have a proper local limit with finite vertex degrees. Note that this assumption also implies $\\alpha_1<1$, i.e. it is not possible that almost all faces are $2$-gons.\n\\item[$\\bullet$]\nThe assumption $\\sum_{j \\geq 1} j^2 \\alpha_j <+\\infty$ means that the \\emph{expected} degree of the root face stays finite in the limit. We do not expect this assumption to be necessary. However, one of the steps of our proof (the \"two-holes argument\" of Section~\\ref{sec_arg_deux_trous}) crucially requires a bound on the tail of the degrees of the faces.\n\\end{itemize}\nFinally, the application associating the weight sequence $\\mathbf{q}$ given by Theorem~\\ref{univ_main_thm} to $\\left( \\theta, \\left( \\alpha_j \\right)_{j \\geq 1}\\right)$ is surjective in the sense that every IBPM $\\mathbb M_{\\mathbf{q}}$ for which the degree of the root face has finite expectation can be obtained as a local limit through Theorem~\\ref{univ_main_thm}.\n\n\\paragraph{The heavy tail case.}\nAlthough we could not remove the assumption $\\sum_{j \\geq 1} j^2 \\alpha_j$ in Theorem~\\ref{univ_main_thm}, most of the steps of the proof do not require this assumption. In particular, we can still obtain the following partial result in the general case.\n\n\\begin{thm}\\label{thm_main_more_general}\nLet $(\\mathbf{f}^{n})_{n \\geq 1}$ be a sequence of face degree sequences, and let $(g_n)$ be a sequence such that $v(\\mathbf{f}^{n}, g_n) \\geq 2$ for all $n \\geq 1$. We assume that $|\\mathbf{f}^n| \\to +\\infty$ when $n \\to +\\infty$ and that $\\frac{f^n_j}{|\\mathbf{f}^n|} \\to \\alpha_j$ for all $j \\geq 1$, where $\\sum_{j \\geq 1} j \\alpha_j=1$.\nWe also assume $\\frac{g_n}{|\\mathbf{f}^n|} \\to \\theta$, where $0 \\leq \\theta < \\frac{1}{2} \\sum_{j \\geq 1} (j-1) \\alpha_j$.\n\nThen the sequence of random maps $\\left( M_{\\mathbf{f}^n, g_n} \\right)_{n \\geq 1}$ is tight for the local topology. Moreover, all its subsequential limits are of the form $\\mathbb M_{\\mathbf{Q}}$, where $\\mathbf{Q}$ is a random Boltzmann weight sequence.\n\\end{thm}\n\n\\paragraph{A parametrization of Infinite Bipartite Boltzmann Planar Maps.}\nAs explained briefly above, we also provide a new parametrization of the Boltzmann weight families $\\mathbf{q}$ associated to an IBPM: instead of directly using the Boltzmann weights $q_j$, we parametrize them according to the law of the degree of the root face.\n\n\\begin{thm}\\label{thm_prametrization_rootface}\nLet $\\boldsymbol{\\alpha}$ be a probability measure on $\\mathbb N^*$. Then the set of IBPM for which the half-degree of the root face has law $\\boldsymbol{\\alpha}$ forms a one parameter family $\\left( \\mathbb M_{\\mathbf{q}^{(\\omega)}} \\right)_{\\omega \\geq 1}$. Moreover, $\\mathbf{q}^{(\\omega)}$ is critical if and only if $\\omega=1$, and the vertex degrees in $\\mathbb M_{\\mathbf{q}^{(\\omega)}}$ go to infinity when $\\omega \\to +\\infty$.\n\\end{thm}\n\nIn particular, the existence of hyperbolic Boltzmann maps with arbitrarily heavy-tailed face degrees answers a question from~\\cite{C-StFlour}, that was not settled by the results of \\cite[Appendix C]{B18these}. Moreover, we can think of $\\mathbf{q}^{(\\omega)}$ as interpolating between a critical non-hyperbolic map, and a degenerate map with infinite vertex degrees.\n\n\\paragraph{Asymptotic enumeration.}\nLike for triangulations, the most natural way to try to prove Theorem~\\ref{univ_main_thm} would be to obtain precise asymptotics on the counting of maps with prescribed genus and face degrees, in order to mimic classical arguments going back to~\\cite{AS03}. However, such asymptotics are not available and seem difficult to obtain. On the other hand, just like in~\\cite{BL19} for triangulations, once Theorem~\\ref{univ_main_thm} is proved, applying the arguments of~\\cite{AS03} \"backwards\" allows to obtain a result about convergence of the ratio when we add one face of fixed degree. We denote by $\\beta_g(\\mathbf{f})$ the number of bipartite maps of genus $g$ with face degrees prescribed by $\\mathbf{f}$.\n\n\\begin{corr}\\label{prop_cv_ratio}\nLet $\\left( \\mathbf{f}^{n} \\right)_{n \\geq 0}$ and $\\left( g_n \\right)_{n \\geq 0}$ be such that $\\frac{g_n}{n}\\to\\theta$ and $\\frac{f^n_j}{|\\mathbf{f}^n|}\\to \\alpha_j$ for all $j \\geq 1$. We assume that $\\sum j \\alpha_j=1$, that $0 \\leq \\theta < \\frac{1}{2} \\sum (j-1) \\alpha_j$ and that $\\sum j^2 \\alpha_j < +\\infty$. We recall that by Theorem~\\ref{univ_main_thm}, there is a weight sequence $\\mathbf{q}$ such that $M_{\\mathbf{f}^{n},g_n}$ converges locally to $\\mathbb{M}_{\\mathbf{q}}$. Then for all $j \\geq 1$, we have\n\\begin{equation}\\label{eq_cv_ratio}\n\\frac{\\beta_{g_n}(\\mathbf{f}^{n}-\\mathbf{1}_j)}{\\beta_{g_n}(\\mathbf{f}^{n})} \\xrightarrow[n \\to +\\infty]{} C_2(\\mathbf{q})q_j.\n\\end{equation}\n\\end{corr}\n\nWe also believe that the following is true:\n\n\\begin{conj}\\label{thm_univ_asympto}\nLet $(\\mathbf{f}^{n})_{n \\geq 1}$ be a sequence of face degree sequences, and let $(g_n)$ be a sequence such that $v(\\mathbf{f}^{n}, g_n) \\geq 2$ for all $n \\geq 1$. We assume that $|\\mathbf{f}^n| \\to +\\infty$ when $n \\to +\\infty$ and that $\\frac{f^n_j}{|\\mathbf{f}^n|} \\to \\alpha_j$ for all $j \\geq 1$, where $\\sum_{j \\geq 1} j \\alpha_j=1$.\nWe also assume $\\frac{g_n}{|\\mathbf{f}^n|} \\to \\theta$, where $0 \\leq \\theta \\leq \\frac{1}{2} \\sum_{j \\geq 1} (j-1) \\alpha_j$.\nFinally, assume that $\\sum_{j \\geq 1} j^2 \\alpha_j <+\\infty$.\nThen\n\\[\\beta_{g_n}(\\mathbf{f}^{n})= |\\mathbf{f}^n|^{2g_n} \\exp \\left( \\varphi \\left( \\theta,\\left( \\alpha_j \\right)_{j \\geq 1} \\right) |\\mathbf{f}^n| + o \\left( |\\mathbf{f}^n| \\right) \\right),\\]\nwhere $\\varphi$ is some function.\n\\end{conj}\n\nMore precisely, in~\\cite{BL19}, the proof consists of first using the analog of Corollary~\\ref{prop_cv_ratio} to estimate the ratio between triangulations with any genus and triangulations with a genus close to maximal (say with $\\varepsilon |\\mathbf{f}|$ vertices). To count such triangulations, we contracted a spanning tree to reduce the problem to triangulations with only one vertex, for which explicit formulas are known. This \"contraction\" is the step that is difficult to extend to our setting here: while for triangulations we simply obtained a triangulation with less faces, here the impact on the face degrees may become much more complex. This is why we leave the question as open.\n\n\\paragraph{Sketch of the proof of Theorem~\\ref{univ_main_thm}: common points and differences with the triangular case.}\nThe proof is a combination of combinatorial and probabilistic ideas. It follows the same global strategy as in \\cite{BL19}, which shows the robustness of the approach of~\\cite{BL19}. However, new difficulties arise at each of the steps, which makes the overall proof much longer. \n\nMore precisely, the first step consists of showing the tightness of $M_{\\mathbf{f}^n, g_n}$. This follows from a \\emph{bounded ratio lemma} (Lemma~\\ref{lem_BRL}), stating that under certain assumptions the ratio $\\frac{\\beta_g \\left(\\mathbf{f}+\\mathbf{1}_j \\right)}{\\beta_g \\left(\\mathbf{f} \\right)}$ is bounded. As in~\\cite{BL19}, this lemma is established using surgery operations to remove a face, but this surgery can affect a larger region than in the triangular case, which makes it more elaborated. The second step is to prove that any subsequential limit is planar and one-ended. This relies on the recurrence proved by the second author in~\\cite{Lo19}, and only requires minor adaptations compared to~\\cite{BL19}. We then notice that any subsequential limit enjoys a weak spatial Markov property, which implies that it must be of the form $\\mathbb M_{\\mathbf{Q}}$, for some random weight sequence $\\mathbf{Q}$. This part is also similar to~\\cite{BL19}, although additional technicalities arise. All these arguments do not use any assumption on the tail of the degrees of the faces, and prove Theorem~\\ref{thm_main_more_general}.\n\nThe end of the proof consists in showing that $\\mathbf{Q}$ is actually deterministic. As in~\\cite{BL19}, this step relies on a surgery argument called the \\emph{two holes argument}, for which we need to explore two pieces of maps with the exact same boundary length. This is where the assumption $\\sum j^2 \\alpha_j<+\\infty$ is crucial: without it, when we explore a piece of map \"face by face\", the perimeter makes large positive jumps and misses too many values. Another major difference with \\cite{BL19} is in the last step, where we match the average degree in finite models (computed with the Euler formula) and in infinite ones. In particular, we need to argue that a weight sequence $\\mathbf{q}$ is determined by the law of the root face of $\\mathbb{M}_{\\mathbf{q}}$ and the average vertex degree. While for triangulations we were able to obtain an explicit formula for the average vertex degree, this is not the case here. Therefore, we need to develop new arguments making use of the local limit results obtained earlier in the paper. This is also the reason why the link between $\\theta$, $\\left( \\alpha_j\\right)$ and $\\mathbf{q}$ in Theorem~\\ref{univ_main_thm} is not explicit.\n\n\\paragraph{Weakly Markovian bipartite maps.}\nJust like in~\\cite{BL19}, the argument showing that a subsequential limit is a mixture of IBPM is a result of independent interest, so we give its statement here. Let $M$ be a random infinite, one-ended, bipartite planar map. We call $M$ \\emph{weakly Markovian} if for any finite map $m$ with one hole, the probability $m \\subset M$ only depends on the perimeter of $m$ and on the family of degrees of its internal faces. We denote by $\\mathcal{Q}_h$ the set of weight sequences $\\mathbf{q}$ for which $\\mathbb{M}_{\\mathbf{q}}$ exists, and by $\\mathcal{Q}_f \\subset \\mathcal{Q}_h$ the set of those $\\mathbf{q}$ for which the expected degree of the root face in $\\mathbb{M}_{\\mathbf{q}}$ is finite (this will be useful to handle the last assumption in Theorem~\\ref{univ_main_thm}).\n\n\\begin{thm}\\label{thm_weak_Markov_general}\nLet $M$ be a weakly Markovian infinite, one-ended, bipartite random planar map. Then there is a random weight sequence $\\mathbf{Q} \\in \\mathcal{Q}_h$ such that $M$ has the same distribution as $\\mathbb M_{\\mathbf{Q}}$. Moreover, if the degree of the root face of $M$ has finite expectation, then $\\mathbf{Q} \\in \\mathcal{Q}_f$ almost surely.\n\\end{thm}\n\n\\paragraph{Structure of the paper.}\nIn Section~\\ref{sec_univ_prelim}, we review basic definitions on maps, and previous combinatorial results that will be used in all the paper. We also introduce the IBPM and describe various parametrizations of the set of IBPMs, and in particular prove Theorem~\\ref{thm_prametrization_rootface}. In Section~\\ref{sec_univ_tight}, under the assumptions of Theorem~\\ref{thm_main_more_general} (i.e. without the assumption on the tail of face degrees), we prove that the maps $M_{\\mathbf{f}^n, g_n}$ are tight for the local topology, and that any subsequential limit is a.s. planar and one-ended. Section~\\ref{sec_univ_markov} is devoted to the proof of Theorem~\\ref{thm_weak_Markov_general}, which implies that any subsequential limit of $M_{\\mathbf{f}^n, g_n}$ is an IBPM with random parameters. This is sufficient to prove Theorem~\\ref{thm_main_more_general}. In Section~\\ref{sec_univ_end}, we conclude the proof of Theorem~\\ref{univ_main_thm} by showing that the parameters are deterministic and depend only on $\\theta$ and $\\left( \\alpha_j \\right)_{j \\geq 1}$. In Section~\\ref{sec_univ_asymp}, we deduce the combinatorial estimate of Corollary~\\ref{prop_cv_ratio} from Theorem~\\ref{univ_main_thm}. Finally, the Appendices contain the proofs of some technical results.\n\n\\tableofcontents\n\n\\newpage\n\n\\section*{Index of notations} \\label{sec_univ_index}\n\n\\addcontentsline{toc}{section}{\\protect\\numberline{}Index of notations}\n\nIn general, we will we will use lower case letters such as $m$ to denote deterministic objects or quantities, upper case letters such as $M$ for random objects and $\\mathtt{mathcal}$ letters such as $\\mathcal{M}$ for sets of objects. We will use $\\mathtt{mathbf}$ characters such as $\\mathbf{q}$ for sequences, and normal characters such as $q_j$ for their terms.\n\n\\begin{itemize}\n\\item\n$\\mathbf{f}=(f_j)_{j \\geq 1}$: denotes a face degree sequence ($\\mathbf{F}$ will denote a random face degree sequence).\n\\item\n$g$: will denote the genus.\n\\item\n$\\mathcal{B}_{g}(\\mathbf{f})$: set of finite bipartite maps with genus $g$ and $f_j$ faces of degree $2j$ for all $j \\geq 1$.\n\\item\n$\\beta_{g}(\\mathbf{f})$: cardinality of $\\mathcal{B}_{g}(\\mathbf{f})$.\n\\item\n$M_{\\mathbf{f},g}$: uniform random map in $\\mathcal{B}_{g}(\\mathbf{f})$.\n\\item\n$|\\mathbf{f}|=\\sum_{j \\geq 1} j f_j$ (i.e. the number of edges of a map in $\\mathcal{B}_{g}(\\mathbf{f})$).\n\\item\n$v(\\mathbf{f}, g) = 2-2g+\\sum_{j \\geq 1} (j-1)f_j$ (i.e. the number of vertices of a map in $\\mathcal{B}_{g}(\\mathbf{f})$).\n\\item\n$\\overline{\\mathcal{B}}$: space of finite or infinite bipartite maps with finite vertex degrees, equipped with the local distance $d_{\\mathrm{loc}}$.\n\\item\n$\\overline{\\mathcal{B}}^*$: space of finite or infinite bipartite maps with finite or infinite vertex degrees, equipped with the dual local distance $d_{\\mathrm{loc}}^*$.\n\\item\n$\\theta$: limit value of $\\frac{g}{|\\mathbf{f}|}$ when $|\\mathbf{f}| \\to +\\infty$.\n\\item\n$\\mathbf{q}=(q_j)_{j \\geq 1}$: denotes a weight sequence ($\\mathbf{Q}$ denotes a random weight sequence).\n\\item\n$\\mathbb M_{\\mathbf{q}}$: the infinite bipartite Boltzmann planar map with weight sequence $\\mathbf{q}$.\n\\item\n$W_p(\\mathbf{q})$: partition function of finite Boltzmann bipartite maps of the $2p$-gon with weights $\\mathbf{q}$.\n\\item\n$\\mathcal{Q}=[0,1]^{\\mathbb N^*}$.\n\\item\n$\\mathcal{Q}^*=\\left\\{ \\mathbf{q}=(q_j)_{j \\geq 1} \\in \\mathcal{Q} | \\exists j \\geq 2, q_j>0 \\right\\}$.\n\\item\n$\\mathcal{Q}_a$: set of admissible families of Boltzmann weights $\\mathbf{q}$, i.e. such that $W_p(\\mathbf{q})<+\\infty$.\n\\item\n$\\mathcal{Q}_h$: set of Boltzmann weights for which $\\mathbb M_{\\mathbf{q}}$ exists. We have $\\mathcal{Q}_h \\subset \\mathcal{Q}_a \\cap \\mathcal{Q}^*$.\n\\item\n$\\mathcal{Q}_f$: set of Boltzmann weights $\\mathbf{q} \\in \\mathcal{Q}_h$ for which the expectation of the degree of the root face of $\\mathbb M_{\\mathbf{q}}$ is finite.\n\\item\n$c_{\\mathbf{q}}$: for $\\mathbf{q} \\in \\mathcal{Q}_a$, denotes the solution of the equation \\[\\sum_{j \\geq 1} q_j \\frac{1}{4^{j-1}} \\binom{2j-1}{j-1} c_{\\mathbf{q}}^{j-1} = 1-\\frac{4}{c_{\\mathbf{q}}}.\\]\n\\item\n$\\nu_{\\mathbf{q}}(i)= \\left\\{ \n\\begin{array}{ll}\nq_{i+1} \\, c_{\\mathbf{q}}^{i} & \\mbox{ if $i \\geq 0$,} \\\\\n2 W_{-1-i}(\\mathbf{q}) \\, c_{\\mathbf{q}}^i & \\mbox{ if $i \\leq -1$.}\n\\end{array} \\right.$. Step distribution of the random walk associated to the perimeter process of a finite $\\mathbf{q}$-Boltzmann planar map.\n\\item\n$\\omega_{\\mathbf{q}} \\geq 1$: for $\\mathbf{q} \\in \\mathcal{Q}_h$, denotes the solution (other than $1$, unless $\\mathbf{q}$ is critical) of \\[\\sum_{i \\in \\mathbb Z} \\omega^i \\nu_{\\mathbf{q}}(i)=1.\\]\n\\item\n$\\widetilde{\\nu}_{\\mathbf{q}}(i)=\\omega_{\\mathbf{q}}^i \\nu_{\\mathbf{q}}(i)$. Step distribution of the random walk on $\\mathbb Z$ associated to the peeling process of $\\mathbb M_{\\mathbf{q}}$.\n\\item\n$C_p(\\mathbf{q})$: for $\\mathbf{q} \\in \\mathcal{Q}_h$ and $p \\geq 1$, constants such that \\[\\P \\left( m \\subset \\mathbb M_{\\mathbf{q}} \\right)=C_p(\\mathbf{q}) \\times \\prod_{f \\in m} q_{\\mathrm{deg}(f)\/2}\\] for every finite map $m$ with one hole of perimeter $2p$.\n\\item\n$h_p(\\omega)=\\sum_{i=0}^{p-1} (4 \\omega)^{-i} \\binom{2i}{i}$. We have $C_p(\\mathbf{q})= (c_{\\mathbf{q}} \\omega_{\\mathbf{q}})^{p-1} h_p(\\omega_{\\mathbf{q}})$. Also, the perimeter process associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}}$ is a Doob transform of the $\\widetilde{\\nu}_{\\mathbf{q}}$-random walk by the harmonic function $\\left( h_p(\\omega_{\\mathbf{q}}) \\right)_{p \\geq 1}$.\n\\item\n$a_j(\\mathbf{q})=\\frac{1}{j} \\P \\left( \\mbox{the degree of the root face of $\\mathbb M_{\\mathbf{q}}$ is $2j$}\\right)$ for all $j \\geq 1$.\n\\item\n$\\alpha_j$: denotes a possible value of $a_j(\\mathbf{q})$, or the limit of the ratio $\\frac{f_j}{|\\mathbf{f}|}$ when $|\\mathbf{f}| \\to +\\infty$. We will always have $\\sum_{j \\geq 1} j \\alpha_j=1$ and $\\alpha_1<1$.\n\\item\n$\\mathbf{q}^{(\\omega)}$: once $\\left( \\alpha_j \\right)_{j \\geq 1}$ has been fixed, denotes the weight sequence for which $\\omega_{\\mathbf{q}}=\\omega$, and the law of the degree of the root face is described by $\\left( \\alpha_j \\right)_{j \\geq 1}$.\n\\item\n$\\mathcal{A}$: denotes a peeling algorithm.\n\\item\n$\\mathcal{E}_t^{\\mathcal{A}}(m)$: explored map after $t$ filled-in peeling steps on the map $m$ using algorithm $\\mathcal{A}$. This is a finite map with holes.\n\\item\n$P_t, V_t$: denote respectively the perimeter (i.e. the half-length of the boundary of the hole) and the volume (i.e. the total number of edges) of the explored map after $t$ steps during a peeling exploration.\n\\item\n$d(\\mathbf{q})=\\mathbb E \\left[ \\left( \\mbox{degree of the root vertex in $\\mathbb M_{\\mathbf{q}}$} \\right)^{-1} \\right]$.\n\\item\n$r_j(\\mathbf{q})=\\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}} \\right)^{j-1} q_j=\\lim_{t \\to +\\infty} \\frac{1}{t} \\sum_{i=0}^{t-1} \\mathbbm{1}_{P_{i+1}-P_i=j-1}$ for $\\mathbf{q} \\in \\mathcal{Q}_h$ and $j \\geq 1$, where $P$ is the perimeter process associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}}$ (see Proposition~\\ref{prop_q_as_limit}).\n\\item\n$r_{\\infty}(\\mathbf{q}) = \\frac{ \\left(\\sqrt{\\omega_{\\mathbf{q}}}-\\sqrt{\\omega_{\\mathbf{q}}-1} \\right)^2}{2 \\sqrt{\\omega_{\\mathbf{q}}(\\omega_{\\mathbf{q}}-1)}} = \\lim_{n \\to +\\infty} \\frac{V_n-2P_n}{n}$ for $\\mathbf{q} \\in \\mathcal{Q}_h$ and $j \\geq 1$, where $P$ and $V$ are the perimeter and volume processes associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}}$ (see Proposition~\\ref{prop_q_as_limit}).\n\\end{itemize}\n\n\\newpage\n\n\\section{Preliminaries}\\label{sec_univ_prelim}\n\nOur purpose in this section is to recall basic definitions related to maps, local topology and peeling explorations as well as combinatorial results from previous works, and to introduce precisely the infinite objects that will appear in this paper.\n\n\\subsection{Definitions: maps and local topology}\n\\label{subsec_univ_defns}\n\\label{subsec_local_topology_univ}\n\n\\paragraph{Maps.} A (finite or infinite) \\emph{map} $M$ is a way to glue a finite or countable collection of finite oriented polygons, called the \\emph{faces}, along their edges in a connected way. By forgetting the faces of $M$ and looking only at its vertices and edges, we obtain a graph $G$ (if $M$ is infinite, then $G$ may have vertices with infinite degree). The maps that we consider will always be \\emph{rooted}, i.e. equipped with a distinguished oriented edge called the \\emph{root edge}. The face on the right of the root edge is the \\emph{root face}, and the vertex at the start of the root edge is the \\emph{root vertex}.\n\nThe \\emph{dual map} of a map $m$ is the map $m^*$ whose vertices are the faces of $m$, and whose edges are the dual edges to those of $m$. We root $m^*$ at the oriented edge crossing the root edge of $m$ from left to right (see Figure~\\ref{fig_map_dual}).\nIf the number of faces is finite, then $M$ is always homeomorphic to an orientable topological surface, so we can define the genus of $M$ as the genus of this surface. In particular, we call a map \\emph{planar} if it has genus $0$.\n\nA \\emph{bipartite map} is a rooted map where it is possible to color every vertex in black or white without any monochromatic edge. By convention, we may assume that the root is always oriented from white to black, and each edge of the map has a natural orientation from white to black. In a bipartite map, all faces have an even degree. In what follows, we will only deal with bipartite maps (except when dealing with dual maps). Therefore, we will not always specify that the map we consider is bipartite.\n\nFor every $\\mathbf{f}=(f_j)_{j \\geq 1}$ and $g\\geq 0$, we will denote by $\\mathcal{B}_g(\\mathbf{f})$ the set of bipartite maps of genus $g$ with exactly $f_j$ faces of degree $2j$ for all $j\\geq 1$.\nA map of $\\mathcal{B}_g(\\mathbf{f})$ has $|\\mathbf{f}| =\\sum_{j\\geq 1} j f_j$ edges, $\\sum_{j\\geq 1} f_j$ faces and $v(\\mathbf{f}, g)=2-2g+\\sum_{j\\geq 1} (j-1)f_j$ vertices by the Euler formula. In particular, such a map exists if and only if $v(\\mathbf{f},g) \\geq 2$. We will denote by $\\beta_g(\\mathbf{f})$ the cardinality of $\\mathcal{B}_g(\\mathbf{f})$.\n\n\\begin{figure}\n\\center\n\\includegraphics[scale=0.5]{carte_duale}\n\\caption{A map (in black) and its dual (in blue). The arrows mark the roots.}\\label{fig_map_dual}\n\\end{figure}\n\n\\paragraph{Maps with boundaries.} We will need to consider two different notions of bipartite maps with boundaries, that we call \\emph{maps with holes} and \\emph{maps of multi-polygons}. Roughly speaking, the first ones will be used to describe a small neighbourhood of the root in a larger map, and the second ones to describe the complement of this neighbourhood. Note that, since we will only consider bipartite maps in this work, we assume in both definitions that the maps are bipartite.\n\n\\begin{defn}\nA \\emph{map with holes} is a finite, bipartite map with a set of marked faces (called \\emph{holes}) such that:\n\\begin{itemize}\n\\item\nthe boundary of each hole is a simple cycle,\n\\item\nthe boundaries of the different holes are vertex-disjoint,\n\\item\nthe adjacency graph of the \\emph{internal faces} (i.e. the faces that are not holes) is connected,\n\\item\nthe root edge may be any oriented edge of the map.\n\\end{itemize}\nBy convention, the map consisting of two vertices joined by a single edge is a map with one hole and no internal face.\nIf $m$ is a map with holes, we denote by $\\partial m$ its boundary, i.e. the union of the boundaries of its holes.\n\\end{defn}\n\n\\begin{defn}\nLet $\\ell \\geq 1$ and $p_1, p_2, \\dots, p_{\\ell} \\geq 1$. A \\emph{map of the $(2p_1, \\dots, 2p_{\\ell})$-gon} is a finite or infinite bipartite map with $\\ell$ marked oriented edges $(e_i)_{1 \\leq i \\leq \\ell}$, such that:\n\\begin{itemize}\n\\item\n$e_1$ is the root edge,\n\\item\nthe faces on the right of the $e_i$ are distinct,\n\\item\nfor all $1 \\leq i \\leq \\ell$, the face on the right of $e_i$ has degree $2p_i$.\n\\end{itemize}\nThe faces on the right of the marked edges are called \\emph{external faces}, and the other ones are called the \\emph{internal faces}.\nWe denote by $\\mathcal{B}^{(p_1,p_2,\\dots,p_{\\ell})}_g(\\mathbf{f})$ the set of bipartite maps of the $(2p_1,2p_2,\\dots,2p_{\\ell})$-gon of genus $g$ with interior faces given by $\\textbf{f}$. We also denote by $\\beta^{(p_1,p_2,\\dots,p_{\\ell})}_g(\\mathbf{f})$ its cardinal, with the convention that $\\beta_g^{(0)}(\\mathbf{f})$ is $1$ if $g=0$ and $\\mathbf{f}=\\mathbf{0}$, and $0$ otherwise.\n\\end{defn}\n\nNote that, in this second definition, we do not ask that the boundaries are simple or disjoint. The convention for the $0$-gon can be interpreted as saying that the only map of the $0$-gon is the map with $1$ vertex, no edge and no internal face.\n\n\n\\paragraph{Map inclusion.}\nGiven a map $m$, let $m^*$ be its dual map. Let $\\mathfrak{e}$ be a finite, connected subset of edges of $m^*$ such that the root vertex of $m^*$ is incident to $\\mathfrak{e}$. To $\\mathfrak{e}$, we associate the map $m_{\\mathfrak{e}}$ that is obtained by gluing the faces of $m$ corresponding to the vertices of $m^*$ incident to $\\mathfrak{e}$ along the dual of the edges of $\\mathfrak{e}$ (see Figure~\\ref{fig_univ_lazy_inclusion}). Note that $m_{\\mathfrak{e}}$, once rooted at the root edge of $m$, is a map with holes. We will refer to $m_{\\mathfrak{e}}$ as the \\emph{submap of $m$ spanned by $\\mathfrak{e}$}.\n\nIf $m'$ is a map with holes and $m$ is a (finite or infinite) map, we write\n\\[m' \\subset m\\]\nif $m'$ can be obtained from $m$ by the procedure described above. By convention, we also write $m' \\subset m$ if $m'$ is the trivial map with two vertices and one edge, or if $m'$ consists of a simple cycle with the same perimeter as the root face of $m$ (which corresponds to the case $\\mathfrak{e} = \\emptyset$).\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=1]{lazy_inclusion}\n\\caption{Inclusion of bipartite maps, on an example. On the right, the map $m$ and, in red, the set of dual edges $\\mathfrak{e}$. On the left, the map $m_{\\mathfrak{e}}$.}\\label{fig_univ_lazy_inclusion}\n\\end{figure}\n\nEquivalently, we have $m' \\subset m$ if $m$ can be obtained from $m'$ by gluing one or several maps of multipolygons in the holes of $m'$. We highlight that this definition of map inclusion is taken from \\cite{C-StFlour} and is tailored for the \\emph{lazy peeling process} of \\cite{Bud15}. More precisely, maps of multipolygons may not have simple nor disjoint boundaries, so if $m' \\subset m$, it is possible that two boundary edges of $m'$ actually coincide in $m$.\n\n\\paragraph{Local convergence and dual local convergence.}\nThe goal of this paragraph is to recall the definition of local convergence in a setting that is not restricted to planar maps. We denote by $\\overline{\\mathcal{B}}$ the set of finite or infinite bipartite maps in which all the vertices have finite degrees. A map $m$ is naturally equipped with a \\emph{graph distance} $d_m$ on the set of its vertices. If $m \\in \\overline{\\mathcal{B}}$, for every $r \\geq 1$, we denote by $B_r(m)$ the submap of $m$ spanned by the duals of those edges of $m$ which have an endpoint at distance $d_m$ at most $r-1$ from the root vertex. The map $B_r(m)$ is then a map with holes. We also write $B_0(m)$ for the trivial bipartite map consisting of only one edge.\n\nFor any two maps $m,m' \\in \\overline{\\mathcal{B}}$, we write\n\\[d_{\\mathrm{loc}}(m,m')=\\left( 1+\\max \\{r \\geq 0 | B_r(m)=B_r(m')\\} \\right)^{-1}.\\]\nThis is the \\emph{local distance} on $\\overline{\\mathcal{B}}$. As in the planar case, the space $\\overline{\\mathcal{B}}$ is a Polish space and is the completion of the space of finite bipartite maps for $d_{\\mathrm{loc}}$. However, this space is not compact, since $B_1(m)$ may take infinitely many values.\n\nIn our tightness argument, it will be more convenient to first work with a weaker notion of convergence which we call the \\emph{dual local convergence}. We denote by $\\overline{\\mathcal{B}}^*$ the set of finite or infinite bipartite maps (regardless of whether vertex degrees are finite or not).\nLet $m \\in \\overline{\\mathcal{B}}^*$, and let $d_{m^*}$ be the graph distance on its dual. For $r \\geq 1$, we denote by $B_r^{*}(m)$ the submap of $m$ spanned by those edges of $m^*$ which are incident to a face of $m$ lying at distance $d_{m^*}$ at most $r-1$ from the root face of $m$. By convention, let also $B_0^*(m)$ be the map consisting of a simple cycle with the same length as the boundary of the root face. Like $B_r(m)$, the \"ball\" $B^*_r(m)$ is a finite map with holes. For any $m,m' \\in \\overline{\\mathcal{B}}^*$, we write\n\\[d_{\\mathrm{loc}}^*(m,m')=\\left( 1+\\max \\{r \\geq 0 | B_r^*(m)=B_r^*(m')\\} \\right)^{-1}.\\]\nWe call $d^*_{\\mathrm{loc}}$ the \\emph{dual local distance}. Then $\\overline{\\mathcal{B}}^*$ is a Polish space for $d_{\\mathrm{loc}}^*$ and is the completion of the set of finite bipartite maps.\n\n\n\n\n\n\n\n\n\nThe reason why we introduced $d_{\\mathrm{loc}}^*$ is that it will be very easy to obtain tightness for this distance. This will allow us to work directly on infinite objects and deduce tightness for $d_{\\mathrm{loc}}$ later. Tightness for $d_{\\mathrm{loc}}^*$ will be deduced from the next result.\n\n\\begin{lem}\\label{lem_tight_degree_in_ball}\nLet $A(\\cdot)$ be a function from $(0,1)$ to $\\mathbb N$ and let $r \\geq 1$. There is a function $A_r(\\cdot)$ from $(0,1)$ to $\\mathbb N$ with the following property. Let $G$ be a stationary (for the simple random walk) random graph such that, for all $\\varepsilon>0$, we have\n\\[ \\P \\left( \\mathrm{deg}_G(\\rho)>A(\\varepsilon) \\right) \\leq \\varepsilon,\\]\nwhere $\\rho$ is the root vertex. Then for all $\\varepsilon>0$, we have\n\\[ \\P \\left( \\max_{x \\in B_r(G)} \\mathrm{deg}_G(x)>A_r(\\varepsilon) \\right) \\leq \\varepsilon,\\]\nwhere $B_r(G)$ is the ball of radius $r$ centered at the root vertex in $G$.\n\\end{lem}\n\n\\begin{proof}\nThis result goes back to~\\cite{AS03}. More precisely, although not stated explicitely as such, it is proved by induction on $r$ in the proof of tightness of uniform triangulations for the local topology~\\cite[Lemma 4.4]{AS03}. See also~\\cite[Theorem 3.1]{BLS13} for a general statement with minimal assumptions.\n\\end{proof}\n\nFrom here, we easily obtain tightness for $d_{\\mathrm{loc}}^*$ in our setting.\n\n\\begin{lem}\\label{lem_easy_dual_convergence}\nLet $\\mathbf{f}^{n}$ be face degree sequences such that $\\frac{1}{\\left| \\mathbf{f}^{n} \\right|} \\sum_{j> A} j f_j^{n}\\rightarrow 0$ as $A \\to +\\infty$ uniformly in $n$, and let $(g_n)$ be any sequence such that $\\mathcal{B}_{g_n}(\\mathbf{f}^{n}) \\ne \\emptyset$ for every $n$. Recall that $M_{\\mathbf{f}^n, g_n}$ is a uniform map in $\\mathcal{B}_{g_n}(\\mathbf{f}^{n})$. Then $(M_{\\mathbf{f}^n, g_n})$ is tight for $d_{\\mathrm{loc}}^*$.\n\\end{lem}\n\n\\begin{proof}\nLet $M_{\\mathbf{f}^n, g_n}^*$ be the dual map of $(M_{\\mathbf{f}^n, g_n})$. Since $M_{\\mathbf{f}^n, g_n}$ is invariant under rerooting at a uniform edge, the probability that the root vertex of $M_{\\mathbf{f}^n, g_n}^*$ has degree $2j$ is equal to $\\frac{j f^{n}_j}{\\left| \\mathbf{f}^{n} \\right|}$. Therefore, it follows from the assumption of the lemma that the root degree of $M_{\\mathbf{f}^n, g_n}^*$ is tight. Moreover $M_{\\mathbf{f}^n, g_n}^*$ is invariant by rerooting along the simple random walk. Therefore, by Lemma~\\ref{lem_tight_degree_in_ball}, for every $r \\geq 1$, the maximal degree in the ball of radius $r$ centered at the root in $M_{\\mathbf{f}^n, g_n}^*$ is tight. This implies tightness for $d_{\\mathrm{loc}}^*$.\n\\end{proof}\n\nFinally, as in \\cite{BL19}, tightness for $d_{\\mathrm{loc}}$ will be deduced from tightnesss for $d^*_{\\mathrm{loc}}$ using the following result (the proof is the same as for triangulations, and is therefore omitted).\n\n\\begin{lem}\\label{lem_dual_convergence_univ}\nLet $(m_n)$ be a sequence of maps of $\\overline{\\mathcal{B}}$. Assume that\n\\[ m_n \\xrightarrow[n \\to +\\infty]{d_{\\mathrm{loc}}^*} m,\\]\nwith $m \\in \\overline{\\mathcal{B}}$ (i.e. with finite vertex degrees). Then $m_n \\to m$ for $d_{\\mathrm{loc}}$ as $n \\to +\\infty$.\n\\end{lem}\n\n\\subsection{The lazy peeling process of bipartite maps}\n\\label{subsec_lazy_peeling}\n\nWe now recall the definition of the \\emph{lazy peeling process} of maps introduced in \\cite{Bud15} (see also \\cite{C-StFlour} for an extensive study). We will make heavy use of this notion in our proofs.\n\nA \\emph{peeling algorithm} is a function $\\mathcal{A}$ that takes as input a finite bipartite map $m$ with holes, and that outputs an edge $\\mathcal{A}(m)$ on $\\partial m$ (i.e. on the boundary of one of the holes). Given an infinite, planar, one-ended bipartite map $m$ and a peeling algorithm $\\mathcal{A}$, we can define an increasing sequence $\\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)_{t \\geq 0}$ of maps with one hole, such that $\\mathcal{E}_t^{\\mathcal{A}}(m) \\subset m$ for every $t$, in the following way. First, the map $\\mathcal{E}_0^{\\mathcal{A}}(m)$ is the trivial map consisting of the root edge only. For every $t \\geq 0$, we call the edge $\\mathcal{A} \\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)$ the \\emph{peeled edge at time $t$}. Let $F_t$ be the face of $m$ on the other side of this peeled edge (i.e. the side incident to a hole in $\\mathcal{E}_t^{\\mathcal{A}}(m)$). There are two possible cases, as summed up on Figure~\\ref{fig_univ_lazy_peeling}:\n\\begin{itemize}\n\\item either $F_t$ doesn't belong to $\\mathcal{E}_t^{\\mathcal{A}}(m)$, and then $\\mathcal{E}_{t+1}^{\\mathcal{A}}(m)$ is the map obtained from $\\mathcal{E}_t^{\\mathcal{A}}(m)$ by gluing a simple face of size $\\mathrm{deg}(F_t)$ along $\\mathcal{A} \\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)$;\n\\item or $F_t$ belongs to $\\mathcal{E}_t^{\\mathcal{A}}(m)$. In that case, by planarity, there exists an edge $e_t \\in \\mathcal{E}_t^{\\mathcal{A}}(m)$ on the same hole as $\\mathcal{A} \\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)$ such that $e_t$ and $\\mathcal{A} \\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)$ are actually identified in $m$. The map $\\mathcal{E}_{t+1}^{\\mathcal{A}}(m)$ is then obtained from $\\mathcal{E}_t^{\\mathcal{A}}(m)$ by gluing $\\mathcal{A} \\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)$ and $e_t$ together and, if this creates a finite hole, by filling it in the same way as in $m$.\n\\end{itemize}\nSuch an exploration is called \\emph{filled-in}, because all the finite holes are filled at each step.\n\nLet us now discuss two different ways to define peeling explorations on finite or nonplanar maps. We first note that, if we do not fill the region in the second case, then the definition of a peeling exploration still makes sense for any finite or infinite map, with the only difference that the explored part may now have several holes. This is what we will call a \\emph{non-filled-in} peeling exploration, and this will only be used briefly in Section \\ref{subsec_planarity}.\n\nFinally, for a finite map $m$, we define a filled-in exploration using the following convention. Assume that the peeled edge at time $t$ is glued to another boundary edge of $\\mathcal{E}^{\\mathcal{A}}_t(m)$ and forms two holes:\n\\begin{itemize}\n\\item\nif these two holes are connected in $m \\backslash \\mathcal{E}^{\\mathcal{A}}_t(m)$ (which may occur if $m$ is not planar), we stop the exploration at time $t$;\n\\item\nif not, we obtain $\\mathcal{E}^{\\mathcal{A}}_{t+1}(m)$ by filling completely the hole which contains the smallest number of edges in $m$.\n\\end{itemize}\nNote that with these conventions, the map $\\mathcal{E}^{\\mathcal{A}}_t(m)$ always have exactly one hole. This definition will be used to compare peeling explorations of finite and infinite maps in Section \\ref{sec_univ_end}. At this point, the local planarity results from Section \\ref{sec_univ_tight} will allow us to assume that with high probability, the explorations are not stopped too early.\n\n\\begin{figure}\n\\center\n\\includegraphics[scale=1]{lazy_peeling}\n\\caption{The lazy peeling on an example. The peeled edge is in red. Either a new face is discovered (center case), or the chosen edge is glued to another boundary edge (right case, the glued edge is in blue and the filled part in pink).}\\label{fig_univ_lazy_peeling}\n\\end{figure}\n\n\\subsection{Combinatorial enumeration}\\label{subsec_prelim_combi}\n\n\\paragraph{Partition functions for Boltzmann planar maps.}\nBefore describing infinite Boltzmann models in detail, we recall well-known enumeration results in the finite, planar case. We write $\\mathcal{Q}=[0,1]^{\\mathbb N^*}$. Fix a sequence $\\mathbf{q} \\in \\mathcal{Q}$. The partition function of bipartite, planar maps of the $2p$-gon with Boltzmann weights $\\mathbf{q}$ is defined as\n\\[W_p(\\mathbf{q})=\\sum_{m} \\prod_{f \\in m} q_{\\mathrm{deg}(f)\/2},\\]\nwhere the sum spans over all planar bipartite maps $m$ of the $2p$-gon, and the product is over internal faces of $m$. We also denote by $W_p^{\\bullet}(\\mathbf{q})$ the \\emph{pointed} partition function, i.e. the sum obtained by multiplying the weight of a map $m$ by its total number of vertices. Note that $W_1(\\mathbf{q})$ can also be interpreted as the partition function of maps of the sphere.\n\nWe recall from \\cite{MM07} the classical necessary and sufficient condition for the finiteness of these partition functions. Given a weight sequence $\\mathbf{q} \\in \\mathcal{Q}$, let \n\\begin{equation*}\nf_{\\mathbf{q}}(x)=\\sum_{j \\geq 1} q_{j}\\binom{2 j-1}{j-1} x^{j-1}.\n\\end{equation*}\nIf the equation \n\\begin{equation}\\label{eq_univ_admissible}\nf_{\\mathbf{q}}(x)=1-\\frac{1}{x}\n\\end{equation}\nhas a positive solution $Z_{\\mathbf{q}}$ we call $\\mathbf{q}$ \\emph{admissible}, and write $c_{\\mathbf{q}}=4Z_{\\mathbf{q}}$. Then by results from \\cite{MM07}, for all $p \\geq 1$, the partition functions $W_p(\\mathbf{q})$ and $W_p^{\\bullet}(\\mathbf{q})$ are finite if and only if $\\mathbf{q}$ is admissible. Moreover, in this case, for $p \\geq 0$, we have\n\\begin{equation}\\label{eqn_exact_pointed_partition_function}\nW_p^{\\bullet}(\\mathbf{q}) = c_{\\mathbf{q}}^p \\times \\frac{1}{4^p} \\binom{2p}{p}.\n\\end{equation}\nIt is also possible to derive simple integral formulas for $W_p(\\mathbf{q})$ in terms of $c_{\\mathbf{q}}$ but this will not be needed here, see \\cite{C-StFlour} for more details. We denote by $\\mathcal{Q}_a$ the set of admissible weight sequences.\n\nFinally, let $\\mathcal{Q}^*$ be the set of those $\\mathbf{q}=(q_j)_{j \\geq 1} \\in \\mathcal{Q}$ for which there exists $j \\geq 2$ such that $q_j>0$ (which ensures $W_p(\\mathbf{q})>0$ for all $p \\geq 1$). For $\\mathbf{q} \\in \\mathcal{Q}^* \\cap \\mathcal{Q}_a$, we define the \\emph{Boltzmann distribution} with weights $\\mathbf{q}$ on finite planar bipartite maps of the $2p$-gon as\n\\[\\P(m)=\\frac{1}{W_p(\\mathbf{q})} \\prod_{f \\in m} q_{\\mathrm{deg}(f)\/2}\\]\nfor all bipartite planar map $m$ of the $2p$-gon.\n\n\\paragraph{A general recursion for bipartite maps.}\nAs in \\cite{BL19}, we are lacking precise asymptotics on the enumeration of maps when both the genus and the size go to infinity. The following recurrence formula, proved in \\cite{Lo19}, will play the same role as the Goulden--Jackson formula for triangulations \\cite{GJ08}. We set the convention $\\beta_g(\\mathbf{0})=0$ for all $g$. Then, for every $g\\geq 0$ and every face degree sequence $\\mathbf{f}$, we have\n\\begin{equation}\\label{rec_biparti_genre_univ}\n\\binom{|\\mathbf{f}|+1}{2}\\beta_g(\\mathbf{f})=\\hspace{-0.5cm}\\sum_{\\substack{\\mathbf{h}^{(1)}+\\mathbf{h}^{(2)}=\\mathbf{f}\\\\g^{(1)}+g^{(2)}+g^*=g}}\\hspace{-0.5cm}(1+|\\mathbf{h}^{(1)}|)\\binom{v \\left( \\mathbf{h}^{(2)},g^{(2)} \\right) }{2g^*+2}\\beta_{g^{(1)}}(\\mathbf{h}^{(1)}) \\beta_{g^{(2)}}(\\mathbf{h}^{(2)})+\\sum_{g^*\\geq 0}\\binom{v \\left( \\mathbf{f},g \\right) +2g^*}{2g^*+2}\\beta_{g-g*}(\\mathbf{f}),\n\\end{equation}\nwhere we recall that $|\\mathbf{f}|=\\sum_{j \\geq 1} j f_j$ and $v(\\mathbf{f},g)=2-2g+\\sum_j (j-1) f_j$ (i.e. it is the number of vertices of a map with face degrees $\\mathbf{f}$ and genus $g$).\n\n\\subsection{Infinite Boltzmann bipartite planar maps}\n\\label{subsec_IBPM}\n\n\\paragraph{Definition of the models.}\nOur goal here is to recall the definition of infinite Boltzmann bipartite planar maps introduced in \\cite[Appendix C]{B18these} (and earlier in \\cite{Bud15} in the critical case). We also refer to \\cite[Chapter 8]{C-StFlour} for some basic properties of these objects that we will state below.\n\nLet $\\mathbf{q}=(q_j)_{j \\geq 1}$ be a sequence of nonnegative real numbers that we will call the \\emph{Boltzmann weights}. A random infinite bipartite planar map $M$ is called $\\mathbf{q}$-Boltzmann if there are constants $\\left( C_p(\\mathbf{q}) \\right)_{p \\geq 1}$ such that, for every finite bipartite map $m$ with one hole of perimeter $2p$, we have\n\\[ \\P \\left( m \\subset M \\right)=C_p(\\mathbf{q}) \\prod_{f \\in m} q_{\\mathrm{deg}(f)\/2},\\]\nwhere the product is over all internal faces of $m$.\n\nWe will see that given $\\mathbf{q}$, such a map does not always exist, but when it does, it is unique, i.e. the constants $C_p(\\mathbf{q})$ are determined by $\\mathbf{q}$, which justifies the notation $C_p(\\mathbf{q})$. More precisely, as noted in \\cite[Appendix C]{B18these}, if a $\\mathbf{q}$-Boltzmann map exists, then the partition function of maps of a $2$-gon with Boltzmann weights $\\mathbf{q}$ must be finite, which is equivalent to the admissibility criterion \\eqref{eq_univ_admissible}. Moreover, with the notation of Section~\\ref{subsec_univ_defns}, we call $\\mathbf{q}$ \\emph{critical} if $f'_{\\mathbf{q}}(Z_{\\mathbf{q}})=\\frac{1}{Z_{\\mathbf{q}}^2}$ and \\emph{subcritical} if this is not the case.\n\nFinally, we define a measure $\\nu_{\\mathbf{q}}$ on $\\mathbb Z$ as follows:\n\\begin{equation}\\label{eqn_defn_nu}\n\\nu_{\\mathbf{q}}(i)= \\left\\{ \n\\begin{array}{ll}\nq_{i+1} \\, c_{\\mathbf{q}}^{i} & \\mbox{ if $i \\geq 0$,} \\\\\n2 W_{-1-i}(\\mathbf{q}) \\, c_{\\mathbf{q}}^i & \\mbox{ if $i \\leq -1$.}\n\\end{array} \\right.\n\\end{equation}\nAs noted in \\cite{Bud15}, this is the step distribution of the random walk on $\\mathbb Z$ describing the evolution of the perimeter of a finite $\\mathbf{q}$-Boltzmann map with a large perimeter (see also \\cite[Chapter 5.1]{C-StFlour}). Then previous results about the existence of $\\mathbf{q}$-IBPM can be summed up as follows.\n\n\\begin{thm}\\label{thm_rappel_these}\n\\begin{enumerate}\n\\item\nIf a $\\mathbf{q}$-IBPM exists, it is unique (in distribution), so we can denote it by $\\mathbb M_{\\mathbf{q}}$.\n\\item\nIf $\\mathbf{q} \\notin \\mathcal{Q}^* \\cap \\mathcal{Q}_a$, then $\\mathbb M_{\\mathbf{q}}$ does not exist.\n\\item\nIf $\\mathbf{q} \\in \\mathcal{Q}^* \\cap \\mathcal{Q}_a$ is critical, then $\\mathbb M_{\\mathbf{q}}$ exists and $C_p(\\mathbf{q})=c_{\\mathbf{q}}^{p-1} \\times \\frac{2p}{4^p} \\binom{2p}{p}$.\n\\item\nIf $\\mathbf{q} \\in \\mathcal{Q}^* \\cap \\mathcal{Q}_a$ is subcritical, then $\\mathbb M_{\\mathbf{q}}$ exists if and only if the equation\n\\begin{equation}\\label{boltzmann_equation_omega}\n\\sum_{i \\in \\mathbb Z} \\nu_{\\mathbf{q}}(i) \\omega^i =1\n\\end{equation}\nhas a solution $\\omega_{\\mathbf{q}}>1$.\n\\item\nIn this case, the solution $\\omega_{\\mathbf{q}}$ is unique and, for every $p \\geq 1$, we have\n\\begin{equation}\\label{boltzmann_formula_cp}\nC_p(\\mathbf{q})= \\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}} \\right)^{p-1} \\sum_{i=0}^{p-1} (4\\omega_{\\mathbf{q}})^{-i} \\binom{2i}{i}.\n\\end{equation}\n\\end{enumerate}\n\\end{thm}\nThe third point is from \\cite{Bud15}, and the others are from \\cite[Appendix C]{B18these}\\footnote{We have fixed a small mistake from \\cite[Appendix C]{B18these}, where $c_{\\mathbf{q}}$ was omitted in the formula for $C_p(\\mathbf{q})$.}. When it exists, we will call the map $\\mathbb M_{\\mathbf{q}}$ the \\emph{$\\mathbf{q}$-IBPM} (for \\emph{Infinite Boltzmann Planar Map)}.\nWe denote by $\\mathcal{Q}_h \\subset \\mathcal{Q}^* \\cap \\mathcal{Q}_a$ the set of weight sequences $\\mathbf{q}$ for which $\\mathbb M_{\\mathbf{q}}$ exists. We also note that the formula for $C_p(\\mathbf{q})$ in the critical case is a particular case of the subcritical one where $\\omega=1$. Since this function will appear many times later, for $\\omega \\geq 1$ and $p \\geq 1$, we write:\n\\begin{equation}\\label{eqn_defn_homega}\nh_p(\\omega)=\\sum_{i=0}^{p-1} (4\\omega)^{-i} \\binom{2i}{i}.\n\\end{equation}\nIn particular, if $\\omega=1$, then $h_p(\\omega)=\\frac{2p}{4^p} \\binom{2p}{p} \\sim \\frac{2}{\\sqrt{\\pi}} \\sqrt{p}$ as $p \\to +\\infty$. If $\\omega>1$, then $h_p(\\omega) \\to \\sqrt{\\frac{\\omega}{\\omega-1}}$ as $p \\to +\\infty$.\n\n\\paragraph{The random walk $\\widetilde{\\nu}_{\\mathbf{q}}$.}\nTo study the $\\mathbf{q}$-IBPM, we define the measure $\\widetilde{\\nu}_{\\mathbf{q}}$ on $\\mathbb Z$ by $\\widetilde{\\nu}_{\\mathbf{q}}(i)=\\omega_{\\mathbf{q}}^i \\nu_{\\mathbf{q}}(i)$, where $\\omega_{\\mathbf{q}}$ is given by \\eqref{boltzmann_equation_omega} if $\\mathbf{q}$ is subcritical, and $\\omega=1$ if $\\mathbf{q}$ is critical. The random walk with step distribution $\\widetilde{\\nu}_{\\mathbf{q}}$ plays an important role when studying $\\mathbb M_{\\mathbf{q}}$. We first note that, if $\\mathbf{q}$ is not critical, then this walk has a positive drift. Indeed, denoting by $F_{\\mathbf{q}}$ the generating function of $\\nu_{\\mathbf{q}}$, we have\n\\[ \\sum_{i \\in \\mathbb Z} i \\, \\widetilde{\\nu}_{\\mathbf{q}}(i) = F'_{\\mathbf{q}}(\\omega_{\\mathbf{q}})>0,\\]\nsince $F_{\\mathbf{q}}$ is convex and takes the value $1$ both at $1$ and at $\\omega_{\\mathbf{q}}>1$. Note also that it is possible that the drift is $+\\infty$.\n\n\\paragraph{Lazy peeling explorations of the $\\mathbf{q}$-IBPM.}\nWe now perform a few computations related to lazy peeling explorations of the $\\mathbf{q}$-IBPM. For this, we fix a peeling algorithm $\\mathcal{A}$, and consider a filled-in exploration of $\\mathbb M_{\\mathbf{q}}$ according to $\\mathcal{A}$. We recall that $\\mathcal{E}_t^{\\mathcal{A}}(\\mathbb M_{\\mathbf{q}})$ is the explored region after $t$ steps, and we denote by $\\left( \\mathcal{F}_t \\right)_{t \\geq 0}$ the filtration generated by this exploration. We denote by $P_t$ (resp. $V_t$) the half-perimeter (resp. total number of edges) of $\\mathcal{E}_t^{\\mathcal{A}}(\\mathbb M_{\\mathbf{q}})$. We will call $P$ and $V$ the \\emph{perimeter and volume processes} associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}}$.\n\nIt follows from the definition of $\\mathbb M_{\\mathbf{q}}$ that $(P_t, V_t)_{t \\geq 0}$ is a Markov chain on $\\mathbb N^2$ and that its law does not depend on the algorithm $\\mathcal{A}$. More precisely $P$ is a Doob transform of the random walk with step distribution $\\widetilde{\\nu}_{\\mathbf{q}}$, i.e. it has the following transitions:\n\\begin{equation}\\label{peeling_transitions}\n\\P \\left( P_{t+1}=P_t+i | \\mathcal{F}_t \\right)= \\widetilde{\\nu}_{\\mathbf{q}}(i) \\frac{h_{P_t+i}(\\omega_{\\mathbf{q}})}{h_{P_t}(\\omega_{\\mathbf{q}})},\n\\end{equation}\nwhere $h_p(\\omega)$ is given by \\eqref{eqn_defn_homega}. As noted in \\cite{C-StFlour}, this implies that $\\left( h_p(\\omega_{\\mathbf{q}}) \\right)_{p \\geq 1}$ is harmonic on $\\{1,2, \\dots\\}$ for the random walk with step distribution $\\widetilde{\\nu}_{\\mathbf{q}}$, and that for $\\mathbf{q}$ subcritical $P$ has the distribution of this random walk, conditioned to stay positive (if $\\mathbf{q}$ is critical, the conditioning is degenerate, but this can still be made sense of).\n\n\\paragraph{IBPM with finite expected degree of the root face.}\nWe denote by $\\mathcal{Q}_f$ the set of $\\mathbf{q} \\in \\mathcal{Q}_h$ such that the degree of the root face of $\\mathbb M_{\\mathbf{q}}$ has finite expectation. Since our Theorem~\\ref{univ_main_thm} only holds if the expected degree of the root face is finite in the limit, we will need to gather a few consequences of this assumption on $\\mathbf{q}$ and the peeling process of $\\mathbb M_{\\mathbf{q}}$. Note that, for all $\\mathbf{q} \\in \\mathcal{Q}_h$, the degree of the root face is determined by the first peeling step on $\\mathbb M_{\\mathbf{q}}$. More precisely, by \\eqref{peeling_transitions}, we have for all $j \\geq 1$:\n\\begin{equation}\\label{eqn_walk_to_rootface}\n\\P \\left( \\mbox{the root face of $\\mathbb M_{\\mathbf{q}}$ has degree $2j$} \\right)=\\frac{h_j(\\omega_{\\mathbf{q}})}{h_1(\\omega_{\\mathbf{q}})} \\, \\widetilde{\\nu}_{\\mathbf{q}}(j-1) = \\frac{h_j(\\omega_{\\mathbf{q}})}{h_1(\\omega_{\\mathbf{q}})} \\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}} \\right)^{j-1} q_j.\n\\end{equation}\nIf $\\mathbf{q}$ is critical, the right hand-side is equivalent to $\\frac{2}{\\sqrt{\\pi}}\\sqrt{j} c_{\\mathbf{q}}^{j-1} q_j$ as $j \\to +\\infty$, so $\\mathbf{q} \\in \\mathcal{Q}_f$ if and only if\n\\begin{equation}\\label{eqn_finite_32_moment}\n\\sum_{j \\geq 1} j^{3\/2} c_{\\mathbf{q}}^j q_j <+\\infty.\n\\end{equation}\nOn the other hand, we recall (see e.g. \\cite[Chapter 5.2]{C-StFlour}) that a critical weight sequence $\\mathbf{q}$ is called \\emph{of type $\\frac{5}{2}$}, or \\emph{critical generic}, if\n\\[ \\sum_{j \\geq 1} (j-1)(j-2) \\binom{2j-1}{j-1} q_j \\left( \\frac{c_{\\mathbf{q}}}{4} \\right)^{j-3} <+\\infty, \\]\nwhich is clearly equivalent to \\eqref{eqn_finite_32_moment}.\nIn the subcritical case, by \\eqref{eqn_walk_to_rootface}, $\\mathbf{q} \\in \\mathcal{Q}_f$ is equivalent to\n\\[\\sum_{j \\geq 1} j \\widetilde{\\nu}_{\\mathbf{q}}(j) <+\\infty,\\] i.e. the drift of $\\widetilde{\\nu}$ is finite. To sum up:\n\\begin{itemize}\n\\item\nIn the critical case, $\\mathbf{q} \\in \\mathcal{Q}_f$ if and only if $\\mathbf{q}$ is critical generic, which means that the perimeter process $(P_n)$ converges to a $3\/2$-stable L\u00e9vy process with no positive jump, conditioned to be positive (see \\cite[Theorem 10.1]{C-StFlour}). This basically means that $\\mathbf{q}$-Boltzmann finite maps for $\\mathbf{q} \\in \\mathcal{Q}_f$ lie in the domain of attraction of the Brownian map \\cite{MM07}.\n\\item\nIn the subcritical case, $\\mathbf{q} \\in \\mathcal{Q}_f$ if and only if the measure $\\widetilde{\\nu}$ has finite expectation. Since the perimeter process $P$ has the law of a $\\widetilde{\\nu}$-random walk conditioned on an event of positive probability, this means that $P$ has linear growth (instead of super-linear if the expectation of $\\widetilde{\\nu}$ was infinite).\n\\end{itemize}\n\n\\subsection{Four ways to describe Boltzmann weights}\n\n\\paragraph{Four parametrizations of $\\mathcal{Q}_h$.}\nIn this work, we will make use of four different \"coordinate systems\" to navigate through the spaces $\\mathcal{Q}_h$ and $\\mathcal{Q}_f$, each with its own advantages. The goal of this section is to define these parametrizations and to establish some relations between them. In particular, we will prove Theorem \\ref{thm_prametrization_rootface}.\n\nOur first coordinate system, already used in the last pages, consists in using directly the Boltzmann weights $q_j$ for $j \\geq 1$. It is the simplest way to define the model $\\mathbb M_{\\mathbf{q}}$ and gives the simplest description of its law.\n\nThe second parametrization we will use is the one given by Proposition~\\ref{prop_q_as_limit} below: we describe $\\mathbf{q}$ by parameters $r_j(\\mathbf{q}) \\in [0,1)$ for $j \\geq 1$ and $r_{\\infty}(\\mathbf{q}) \\in (0,+\\infty]$. Here $r_j(\\mathbf{q})$ describes the proportion of peeling steps where we discover a face of degree $2j$ during a peeling exploration of $\\mathbb M_{\\mathbf{q}}$, and $r_{\\infty}(\\mathbf{q})$ comes from a comparison between the volume and perimeter growths. The advantage of these parameters is that they allow to \"read\" $\\mathbf{q}$ as an almost sure observable on a peeling exploration of the map $\\mathbb M_{\\mathbf{q}}$. This will be useful in Section~\\ref{sec_arg_deux_trous}.\n\nThe third parametrization consists in using on the one hand the law of the root face, and on the other hand the average degree of the vertices. More precisely, for $j \\geq 1$, we write\n\\[a_j(\\mathbf{q})=\\frac{1}{j} \\P \\left( \\mbox{the root face of $\\mathbb M_{\\mathbf{q}}$ has degree $2j$} \\right).\\]\nWe note that $\\sum_{j \\geq 1} j a_j(\\mathbf{q})=1$ and that $a_1(\\mathbf{q})<1$ since a map consisting only of $2$-gons would have vertices with infinite degrees.\nWe also write $d(\\mathbf{q})=\\mathbb E \\left[ \\frac{1}{\\mathrm{deg}_{\\mathbb M_{\\mathbf{q}}}(\\rho)} \\right]$, where $\\rho$ is the root vertex. The advantage of this parametrization is that the analogues of these quantities are easy to compute if we replace $\\mathbb M_{\\mathbf{q}}$ by a finite uniform map with prescribed genus and face degrees. These parameters are our only way to link the finite and infinite models, and will therefore be useful in the end of the proof of Theorem~\\ref{univ_main_thm}. However, it is not obvious at all that $\\left( a_j(\\mathbf{q}) \\right)_{j \\geq 1}$ and $d(\\mathbf{q})$ are sufficient to characterize $\\mathbf{q}$. We will actually prove this in the end of the paper, only for $\\mathbf{q} \\in \\mathcal{Q}_f$, as a consequence of local convergence arguments (Proposition~\\ref{prop_monotonicity_deg}). \n\nFinally, the fourth coordinate system is the one from Theorem \\ref{thm_prametrization_rootface}: it is a variant of the third one where we replace $d(\\mathbf{q})$ by $\\omega_{\\mathbf{q}}$, which makes it easier to handle. This one is useful as an intermediate step towards the third one. Moreover, contrary to the third one, we can prove rather quickly (Theorem \\ref{thm_prametrization_rootface}) that it provides a nice parametrization of the whole space $\\mathcal{Q}_h$.\n\n\\paragraph{Recovering $\\mathbf{q}$ from explorations of $\\mathbb M_{\\mathbf{q}}$.}\nWe now describe more precisely our second parametrization of $\\mathcal{Q}_h$. The next result basically states that we can recover the weight sequence $\\mathbf{q}$ by just observing the perimeter and volume processes defined above (we recall that the volume is measured by the total number of edges).\n\n\\begin{prop}\\label{prop_q_as_limit}\nLet $\\mathbf{q} \\in \\mathcal{Q}_h$, and let $P$ and $V$ be the perimeter and volume processes associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}}$. We have the following almost sure convergences:\n\\begin{equation}\\label{limit_number_jgons}\n\\frac{1}{t} \\sum_{i=0}^{t-1} \\mathbbm{1}_{P_{i+1}-P_i=j-1} \\xrightarrow[t \\to +\\infty]{a.s.} \\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}} \\right)^{j-1} q_j =: r_j(\\mathbf{q}) \\in [0,1)\n\\end{equation}\nfor every $j \\geq 1$, and\n\\begin{equation}\\label{limit_volume_growth}\n\\frac{V_t-2P_t}{t} \\xrightarrow[t \\to +\\infty]{a.s.} \\frac{ \\left(\\sqrt{\\omega_{\\mathbf{q}}}-\\sqrt{\\omega_{\\mathbf{q}}-1} \\right)^2}{2 \\sqrt{\\omega_{\\mathbf{q}}(\\omega_{\\mathbf{q}}-1)}} =: r_{\\infty}(\\mathbf{q}) \\in (0,+\\infty].\n\\end{equation}\nMoreover, the weight sequence $\\mathbf{q}$ is a measurable function of the numbers $r_j(\\mathbf{q})$ for $j \\in \\mathbb N^* \\cup \\{\\infty\\}$.\n\\end{prop}\n\n\\begin{proof}\nIn the subcritical case, the second convergence is Proposition 10.12 of \\cite{C-StFlour}. In the critical case, we have $\\omega_{\\mathbf{q}}=1$ so the right-hand side of \\eqref{limit_volume_growth} is infinite, and the result follows from Lemma 10.9 of \\cite{C-StFlour}.\n\nLet us now prove the first convergence. For this, we first note that we have $P_t \\to +\\infty$ almost surely as $t \\to +\\infty$. Indeed, this again follows from \\cite[Proposition 10.12]{C-StFlour} in the subcritical case and from \\cite[Lemma 10.9]{C-StFlour} in the critical case. On the other hand, given the asymptotics for $h_p(\\omega)$ right after \\eqref{eqn_defn_homega}, for any fixed $j \\geq 1$, we have $\\frac{h_{p+j-1}(\\omega_{\\mathbf{q}})}{h_p(\\omega_{\\mathbf{q}})} \\to 1$ as $p \\to +\\infty$. It follows that\n\\[ \\P \\left( P_{t+1}-P_t=j-1 | \\mathcal{F}_t \\right) \\xrightarrow[t \\to +\\infty]{a.s.} \\widetilde{\\nu}_{\\mathbf{q}}(j-1) = \\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}} \\right)^{j-1} q_j, \\]\nand the first convergence follows by the law of large numbers.\n\nFinally, the function $\\omega \\to \\frac{ \\left(\\sqrt{\\omega}-\\sqrt{\\omega-1} \\right)^2}{2 \\sqrt{\\omega(\\omega-1)}}$ is a decreasing homeomorphism from $[1,+\\infty)$ to $(0,+\\infty]$, so $\\omega_{\\mathbf{q}}$ is a measurable function of $r_{\\infty}(\\mathbf{q})$. Moreover, by the\ndefinition of $c_{\\mathbf{q}}$~\\eqref{eq_univ_admissible}, we have\n\\[ 1-\\frac{4}{c_{\\mathbf{q}}} = \\sum_{j \\geq 1} \\binom{2j-1}{j-1} q_j \\left( \\frac{c_{\\mathbf{q}}}{4} \\right)^{j-1} = \\sum_{j \\geq 1} \\frac{1}{4^j} \\binom{2j-1}{j-1} \\frac{r_j(\\mathbf{q})}{\\omega_{\\mathbf{q}}^{j-1}},\\]\nwhich implies that $c_{\\mathbf{q}}$ is a measurable function of $\\omega_{\\mathbf{q}}$ and the numbers $r_j(\\mathbf{q})$ for $j \\in \\mathbb N^*$. Finally, given $c_{\\mathbf{q}}$ and the $r_j(\\mathbf{q})$, we easily recover the $q_j$ from~\\eqref{limit_number_jgons}.\n\\end{proof}\n\n\\paragraph{Weight sequences corresponding to a given distribution of the root face.}\nWe now prove Theorem~\\ref{thm_prametrization_rootface} by showing that our fourth parametrization is indeed bijective. We first state the precise version of Theorem~\\ref{thm_prametrization_rootface}. We recall that for $\\mathbf{q} \\in \\mathcal{Q}_h$, the numbers $a_j(\\mathbf{q})$ satisfy $\\sum_{j \\geq 1} j a_j(\\mathbf{q})=1$ and $\\alpha_1<1$, and we have $\\omega_{\\mathbf{q}} \\geq 1$.\n\n\\begin{prop}\\label{prop_third_parametrization}\nLet $(\\alpha_j)_{j \\geq 1}$ be such that $\\sum_{j \\geq 1} j \\alpha_j=1$ and $\\alpha_1 < 1$, and let $\\omega \\geq 1$. Then there is a unique $\\mathbf{q} \\in \\mathcal{Q}_h$ such that\n\\[ \\omega_{\\mathbf{q}}=\\omega \\mbox{ and } \\forall j \\geq 1, \\, a_j(\\mathbf{q})=\\alpha_j.\\]\nMoreover, this weight sequence $\\mathbf{q}$ is given by\n\\begin{equation}\\label{eqn_qjomega_univ}\nq_j=\\frac{j \\alpha_j}{\\omega^{j-1}h_{\\omega}(j)} \\left( \\frac{1-\\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} \\frac{i \\alpha_i}{\\omega^{i-1} h_{\\omega}(i)} }{4} \\right)^{j-1}.\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_third_parametrization}]\nWe start with uniqueness. We note that\n\\begin{equation}\\label{eqn_alpha_from_q}\na_j(\\mathbf{q})=\\frac{1}{j} C_j(\\mathbf{q}) q_j = \\frac{1}{j} \\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}}\\right)^{j-1} h_{j}(\\omega_{\\mathbf{q}}) q_j,\n\\end{equation}\nso $q_j$ can be obtained as a function of $a_j(\\mathbf{q})=\\alpha_j$, $\\omega_{\\mathbf{q}}$ and $c_{\\mathbf{q}}$. Moreover, by the definition~\\eqref{eq_univ_admissible} of $c_{\\mathbf{q}}$, we have\n\\begin{equation}\\label{eqn_g_function_of_aj}\n1-\\frac{4}{c_{\\mathbf{q}}}=\\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} q_i c_{\\mathbf{q}}^{i-1} = \\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} \\frac{i a_i(\\mathbf{q})}{\\omega_{\\mathbf{q}}^{i-1} h_{i}(\\omega_{\\mathbf{q}})},\n\\end{equation}\nso $c_{\\mathbf{q}}$, and therefore $q_j$ for all $j \\geq 1$, can be deduced from $\\omega_{\\mathbf{q}}$ and $\\left( a_j(\\mathbf{q}) \\right)_{j \\geq 1}$. More precisely, we obtain the formula~\\eqref{eqn_qjomega_univ}, which in particular proves the uniqueness.\n\nTo prove the existence, it is enough to check that, for all $\\omega \\geq 1$ and $(\\alpha_j)_{j \\geq 1}$ with $\\sum j \\alpha_j=1$ and $\\alpha_1<1$, the sequence $\\mathbf{q}$ given by \\eqref{eqn_qjomega_univ} is indeed in $\\mathcal{Q}_h$, with $\\omega_{\\mathbf{q}}=\\omega$ and $a_j(\\mathbf{q})=\\alpha_j$ for all $j$. \nFollowing \\eqref{eqn_g_function_of_aj}, we first write\n\\begin{equation}\\label{eqn_c_of_alpha_omega}\nc=\\frac{4}{1-\\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} \\frac{i \\alpha_i}{\\omega^{i-1} h_{i}(\\omega) }},\n\\end{equation}\nand check that $\\mathbf{q}$ is admissible with $c_{\\mathbf{q}}=c$. First $\\omega^{i-1} h_{i}(\\omega)$ is a polynomial in $\\omega$ with nonnegative coefficients so $\\omega^{i-1} h_{i}(\\omega) \\geq h_i(1)=\\frac{2i}{4^i} \\binom{2i}{i}$. From here, we get\n\\[ \\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} \\frac{i \\alpha_i}{\\omega^{i-1} h_{i}(\\omega) } \\leq \\sum_{i \\geq 1} \\alpha_i < \\sum_{i \\geq 1} i \\alpha_i = 1 \\]\nbecause $\\alpha_1 <1$. Therefore, the numbers $q_j$ are nonnegative and $c>0$, and we can rewrite \\eqref{eqn_qjomega_univ} as\n\\[q_j=\\frac{j \\alpha_j}{(\\omega c)^{j-1} h_{j}(\\omega)}. \\]\nFrom here, we get\n\\[ \\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} q_i c^{i-1} = 1-\\frac{4}{c}\\]\nimmediately by the definition of $c$, which proves $\\mathbf{q} \\in \\mathcal{Q}_a$ and $c_{\\mathbf{q}}=c$.\nAlso, we know that $\\alpha_1<1$ so there is $j \\geq 2$ with $\\alpha_j>0$, which implies $q_j>0$, so $\\mathbf{q} \\in \\mathcal{Q}^*$.\n\nWe now prove $\\mathbf{q} \\in \\mathcal{Q}_h$ with $\\omega_{\\mathbf{q}}=\\omega$, which is equivalent to proving\n\\[ \\sum_{i \\in \\mathbb Z} \\nu_{\\mathbf{q}}(i) \\omega^i = 1,\\]\nwhere we recall that $\\nu_{\\mathbf{q}}$ is defined by \\eqref{eqn_defn_nu}. For this,\ninspired by similar arguments in the critical case (see e.g. \\cite[Lemma 5.2]{C-StFlour}), the basic idea will be to show that $\\left( \\omega^i h_{i}(\\omega) \\right)_{i \\geq 1}$ is harmonic for $\\nu_{\\mathbf{q}}$. More precisely, the equality $\\sum_{i \\geq 1} i \\alpha_i=1$ can be interpreted as a harmonicity relation at $1$: setting $h_i(\\omega)=0$ for $i \\leq -1$, we have\n\\begin{equation}\\label{eqn_harmo_at_one}\n\\sum_{i \\in \\mathbb{Z}} h_{i+1}(\\omega) \\omega^i \\nu_{\\mathbf{q}}(i) = \\sum_{j \\geq 1} \\omega^{j-1} h_{j}(\\omega) c^{j-1} q_j = \\sum_{j \\geq 1} j \\alpha_j = 1 = h_{1}(\\omega),\n\\end{equation}\nwhere in the beginning we do the change of variables $j=i+1$. On the other hand, we know that $h_{p}(\\omega)=\\sum_{i=0}^{p-1} \\omega^{-i} u(i)$, where $u(i)=\\frac{1}{4^i} \\binom{2i}{i}$ for $i \\geq 0$ (and we set the convention $u(i)=0$ for $i \\leq -1$). But the same function $u$ plays an important role in the description of the law of the peeling process of finite Boltzmann maps. In particular, we know that $u$ is $\\nu_{\\mathbf{q}}$-harmonic on positive integers for any admissible weight sequence $\\mathbf{q}$ (this can be found in the proof of Lemma 5.2 in \\cite{C-StFlour}). That is, for all $j \\geq 1$, we have\n\\[ u(j)=\\sum_{i \\in \\mathbb{Z}} \\nu_{\\mathbf{q}}(i) u(i+j).\\]\nMultiplying by $\\omega^{-j}$ and summing over $1 \\leq j \\leq p-1$, we get, for all $p \\geq 1$:\n\\[ h_{p}(\\omega)-h_{1}(\\omega) = \\sum_{i \\in \\mathbb Z} \\omega^i \\nu_{\\mathbf{q}}(i) \\left( h_{p+i}(\\omega) - h_{i+1}(\\omega) \\right). \\]\nSumming this with \\eqref{eqn_harmo_at_one} and dividing by $h_{p}(\\omega)$, we obtain\n\\[ \\sum_{i \\in \\mathbb Z} \\omega^i \\nu_{\\mathbf{q}}(i) \\frac{h_{p+i}(\\omega)}{h_{p}(\\omega)} =1. \\]\nfor all $p \\geq 1$. When $p \\to +\\infty$, we have that\n$h_{p}(\\omega)$ has a positive limit if $\\omega>1$ and is equivalent to $\\frac{2}{\\sqrt{\\pi}}\\sqrt{p}$ if $\\omega=1$, so $\\frac{h_{p+i}(\\omega)}{h_{p}(\\omega)} \\to 1$ in every case. Therefore, by dominated convergence, we get\n\\[ \\sum_{i \\in \\mathbb Z} \\nu_{\\mathbf{q}}(i) \\omega^i=1,\\]\nwhere the domination $\\sum_{i \\in \\mathbb Z} \\nu_{\\mathbf{q}}(i) \\omega^i <+\\infty$ is immediate for negative values of $i$ since $\\omega \\geq 1$, and comes from the convergence of the sum \\eqref{eqn_harmo_at_one} for positive values of $i$. This proves $\\mathbf{q} \\in \\mathcal{Q}_h$ with $\\omega_{\\mathbf{q}}=\\omega$, and from here $a_j(\\mathbf{q})=\\alpha_j$ is immediate using \\eqref{eqn_alpha_from_q}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm_prametrization_rootface}]\nIt is clear from Proposition \\ref{prop_third_parametrization} that weight sequences with a given root face distribution are parametrized by $\\omega \\in [1,+\\infty)$. We denote by $\\mathbf{q}^{(\\omega)}$ the unique weight sequence for which the law of the root face is given by $(\\alpha_j)_{j \\geq 1}$ and for which $\\omega_{\\mathbf{q}^{(\\omega)}}=\\omega$. Then $q^{(1)}$ is critical by definition. Moreover, using \\eqref{eqn_qjomega_univ} and \\eqref{eqn_c_of_alpha_omega}, we get for $i \\geq 0$:\n\\begin{equation}\\label{eqn_omega_infinite}\n\\widetilde{\\nu}_{\\mathbf{q}^{(\\omega)}}(i)=q^{(\\omega)}_{i+1} \\omega^i c_{\\mathbf{q}^{(\\omega)}}^i \\xrightarrow[\\omega \\to +\\infty]{} (i+1) \\alpha_{i+1}.\n\\end{equation}\nThe sum over $i \\geq 0$ of the right-hand side is equal to $1$, so $\\widetilde{\\nu}_{\\mathbf{q}^{(\\omega)}} \\left( (-\\infty,-1] \\right) \\to 0$ as $\\omega \\to +\\infty$. By \\eqref{peeling_transitions}, this means that the probability of peeling cases decreasing the perimeter goes to $0$. Since these are the cases creating cycles in the dual, the dual of $\\mathbb M_{\\mathbf{q}^{(\\omega)}}$ becomes close to a tree when $\\omega \\to +\\infty$, and the vertex degrees in $\\mathbb M_{\\mathbf{q}^{(\\omega)}}$ go to infinity.\n\\end{proof}\n\n\\paragraph{Two technical results on the dependance in $\\omega$.}\nWe conclude this section with two technical results that we will need in the end of the proof (Section~\\ref{subsec_last_step}). Both deal with the way that some quantities depend on the parameter $\\omega$. We fix $(\\alpha_j)_{j \\geq 1}$ such that $\\sum_{j \\geq 1} j \\alpha_j = 1$ and $\\alpha_1<1$. By Proposition~\\ref{prop_third_parametrization}, we can denote by $\\mathbf{q}^{(\\omega)}$ the unique weight sequence for which the law of the root face is given by $(\\alpha_j)_{j \\geq 1}$ and $\\omega_{\\mathbf{q}^{(\\omega)}}=\\omega$.\n\nThe first technical lemma states that we can recover $\\mathbf{q}$ from the law of the root face $(\\alpha_j)_{j \\geq 1}$ and a single weight $q_j$, provided $j \\geq 2$.\n\\begin{lem}\\label{lem_qj_is_monotone}\nFor every $j \\geq 1$, the function $\\omega \\to q_j^{(\\omega)}$ is nonincreasing. Moreover, if $j \\geq 2$ and $\\alpha_j>0$, this function is decreasing.\n\\end{lem}\nSince the proof is not particularly enlightening, we postpone it to Appendix \\ref{appendix_qj_monotone}.\n\nOur second technical lemma is a reinforcement of a part of Proposition~\\ref{prop_q_as_limit} above. It states that the second convergence result~\\eqref{limit_volume_growth} is uniform in $\\omega$ as long as $\\omega$ is bounded away from $1$ and $+\\infty$.\n\n\\begin{lem}\\label{lem_unif_volume}\nLet $\\left( P_t^{(\\omega)} \\right)_{t \\geq 0}$ and $\\left( V_t^{(\\omega)} \\right)_{t \\geq 0}$ denote respectively the perimeter and volume processes associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}^{(\\omega)}}$.\nThe convergence in probability\n\\[ \\frac{V_t^{(\\omega)}-2P_t^{(\\omega)}}{t} \\xrightarrow[t \\to +\\infty]{P)} \\frac{ \\left(\\sqrt{\\omega}-\\sqrt{\\omega-1} \\right)^2}{2 \\sqrt{\\omega(\\omega-1)}} \\]\nis uniform in $\\omega$ over any compact subset $K$ of $(1,+\\infty)$ in the sense that for all $\\varepsilon>0$, there is $t_0>0$ such that, for all $t \\geq t_0$ and $\\omega \\in K$:\n\\[ \\P \\left( \\left| \\frac{V_t^{(\\omega)}-2P_t^{(\\omega)}}{t} - \\frac{ \\left(\\sqrt{\\omega}-\\sqrt{\\omega-1} \\right)^2}{2 \\sqrt{\\omega(\\omega-1)}} \\right| >\\varepsilon \\right) < \\varepsilon. \\]\n\\end{lem}\n\nThe proof of Lemma~\\ref{lem_unif_volume} is an adaptation of the proof of~\\eqref{limit_volume_growth} in~\\cite{C-StFlour}, but using a uniform weak law of large numbers. It is delayed to Appendix~\\ref{subsec_unif_volume}.\n\n\\section{Tightness, planarity and one-endedness}\\label{sec_univ_tight}\n\nIn all this section, we will work in the general setting of Theorem \\ref{thm_main_more_general}, i.e. we do not assume $\\sum_{j \\geq 1} j^2 \\alpha_j<+\\infty$.\n\n\\begin{prop}\\label{prop_tightness_dloc_univ}\nLet $(\\mathbf{f}^n, g_n)_{n \\geq 1}$ be as in Theorem \\ref{thm_main_more_general}. Then the sequence $\\left( M_{\\mathbf{f}^n, g_n} \\right)_{n \\geq 1}$ is tight for $d_{\\mathrm{loc}}$, and every subsequential limit is a.s. planar and one-ended.\n\\end{prop}\n\nOur strategy to prove Proposition~\\ref{prop_tightness_dloc_univ} will be similar to \\cite{BL19}, and in particular relies on a Bounded ratio Lemma (Lemma \\ref{lem_BRL}). Sections \\ref{subsec_BRL}, \\ref{subsec_good_sets} and \\ref{subsec_proof_BRL} are devoted to the proof of the Bounded ratio Lemma, which is significantly more complicated than in \\cite{BL19}. In Section \\ref{subsec_planarity}, we prove that any subsequential limit of $\\left( M_{\\mathbf{f}^n, g_n} \\right)_{n \\geq 1}$ for $d^*_{\\mathrm{loc}}$ (which exist by Lemma~\\ref{lem_easy_dual_convergence}) is planar and one-ended. Finally, in Section \\ref{subsec_finite_degrees}, we finish the proof of Proposition \\ref{prop_tightness_dloc_univ} using Lemma \\ref{lem_dual_convergence_univ} and the Bounded ratio Lemma.\n\n\\subsection{The Bounded ratio Lemma}\n\\label{subsec_BRL}\n\nThe Bounded ratio Lemma below means that, as long as the faces are not too large and the number of vertices remains proportional to the number of edges (i.e. basically under the assumptions of Theorem~\\ref{thm_main_more_general}), removing a face of degree $2j_0$ changes the number of maps by at most a constant factor, provided the faces of degree $2j_0$ represent a positive proportion of the faces. We recall from \\eqref{defn_v_f_g} that $|\\mathbf{f}|$ and $v(\\mathbf{f},g)$ are respectively the number of edges and of vertices of a map with genus $g$ and face degrees given by $\\mathbf{f}$. For $j \\geq 1$, we denote by $\\mathbf{1}_j$ the face degree sequence consisting of a single face of degree $2j$, i.e. $\\left( \\mathbf{1}_j \\right)_i$ is $1$ if $i=j$ and $0$ otherwise.\n\n\\begin{lem}[Bounded ratio Lemma]\\label{lem_BRL}\nWe fix $\\kappa, \\delta>0$ and a function $A:(0,1]\\rightarrow \\mathbb N$. Let $\\mathbf{f}$ be a face degree sequence, and let $g \\geq 0$. We assume that\n\\begin{equation}\\label{eq_petites_faces}\nv(\\mathbf{f}, g) > \\kappa |\\mathbf{f}| \\quad \\mbox{ and } \\quad \\forall \\varepsilon>0, \\, \\sum_{i>A(\\varepsilon)} i f_i< \\varepsilon |\\mathbf{f}|.\n\\end{equation}\nLet also $j_0 \\geq 1$ be such that $j_0 f_{j_0}>\\delta |\\mathbf{f}|$. Then the ratio\n\\[\\frac{\\beta_g(\\mathbf{f})}{\\beta_g(\\mathbf{f}-\\mathbf{1}_{j_0})}\\]\nis bounded by a constant depending only on $\\delta, \\kappa$ and the function $A$.\n\\end{lem}\n\nWe will not try to obtain an explicit constant. As in \\cite{BL19}, we will use the Bounded ratio Lemma to estimate the probability of certain events during peeling explorations, so we will need versions with a boundary. Here are the precise versions that we will need later in the paper.\n\n\\begin{corr}\\label{lem_BRL_boundaries}\nLet $\\kappa, \\delta>0$ and $A(\\cdot)$ be as in Lemma \\ref{lem_BRL}. Then there is a constant $C$ such that the following holds.\n\\begin{enumerate}\n\\item\nLet $p,p',j\\geq 1$. Then there is $N$ such that, for all $\\mathbf{f}$ and $g$ satisfying \\eqref{eq_petites_faces} and $j f_j > \\delta |\\mathbf{f}|$ and $|\\mathbf{f}|>N$, we have\n\\[\\frac{\\beta_g^{(p,p')}(\\mathbf{f})}{\\beta_g^{(p,p')}(\\mathbf{f}-\\mathbf{1}_j)}< C\\]\nand in particular\n\\begin{equation}\\label{eqn_BRL_one_boundary}\n\\frac{\\beta_g^{(p)}(\\mathbf{f})}{\\beta_g^{(p)}(\\mathbf{f}-\\mathbf{1}_j)}< 2C.\n\\end{equation}\n\\item\nLet $p_1, p_2 \\geq 1$ and $i_1, i_2 \\geq 0$. Then there is $N$ such that, for all $\\mathbf{f}$ and $g$ satisfying \\eqref{eq_petites_faces} and $|\\mathbf{f}|>N$, we have\n\\[ \\frac{\\beta_g^{(p_1+i_1,p_2+i_2)}(\\mathbf{f})}{\\beta_g^{(p_1,p_2)}(\\mathbf{f})} < C^{i_1+i_2}. \\]\n\\end{enumerate}\n\\end{corr}\n\nSince this will be very important later, we highlight that the constant $C$ does not depend on $p, p', j$ but that $N$ does. The inequality~\\eqref{eqn_BRL_one_boundary} will be used in the tightness argument just like in \\cite{BL19}, whereas the statements with two boundaries will be needed in the two hole argument (Section \\ref{subsec_same_perimeter}).\n\n\\begin{proof}\nWe first claim that we have the identity\n\\begin{equation}\\label{eqn_add_boundary}\n\\beta_g^{(p,p')}(\\mathbf{f})=\\frac{2p(f_{p}+1)p'(f_{p'}+\\mathbbm{1}_{p \\ne p'})}{2(|\\mathbf{f}|+p+p')}\\beta_g(\\mathbf{f}+\\mathbf{1}_{p}+\\mathbf{1}_{p'}).\n\\end{equation}\nIndeed, the factor $p(f_p+1)$ corresponds to the number of ways to add a second root to a map of $\\mathcal{B}_g(\\mathbf{f}+\\mathbf{1}_p)$ such that this second root has a face of degree $2p$ on its right (with respect to the canonical white to black orientation of edges). The factor $p'(f_{p'}+\\mathbbm{1}_{p \\ne p'})$ corresponds to the number of ways of adding a third root next to a face of degree $2p'$ so that the two root faces are distinct. The $(|\\mathbf{f}|+p+p')$ in the denominator corresponds to forgetting the original root. Moreover, if $(\\mathbf{f},g)$ satisfy the assumptions of Lemma \\ref{lem_BRL} for $\\delta, \\kappa, A(\\cdot)$ and $|\\mathbf{f}|$ is large enough, then $(\\mathbf{f}+\\mathbf{1}_p+\\mathbf{1}_{p'},g)$ also satisfies the assumptions of Lemma \\ref{lem_BRL} for $\\frac{\\delta}{2}, \\frac{\\kappa}{2}, A \\left( \\frac{\\cdot}{2}\\right)$. Therefore, the first point of the corollary follows from Lemma \\ref{lem_BRL} and \\eqref{eqn_add_boundary}. To deduce \\eqref{eqn_BRL_one_boundary}, just take $p'=1$ and use the identity $\\beta_g^{(p,1)}(\\mathbf{f})=|\\mathbf{f}|\\beta_g^{(p)}(\\mathbf{f})$ (adding a $2$-gon is equivalent to marking an edge) and the fact that $|\\mathbf{f}|$ is large enough.\n\nFor the second point, we first note that it is sufficient to prove it for $\\{i_1, i_2\\}=\\{0,1\\}$. Since $C$ does not depend on $(p_1, p_2)$, the general case easily follows by induction on $i_1+i_2$. Without loss of generality, we assume $i_1=1, i_2=0$.\n\nWe now note that there is $\\delta, j_1>0$ depending only on $A(\\cdot)$ such that, if \\eqref{eq_petites_faces} is satisfied, then there is $2 \\leq j \\leq j_1$ such that $jf_j > \\delta |\\mathbf{f}|$ (we can assume $j \\geq 2$ because if there are too many $2$-gons, then the number of vertices cannot be macroscopic). We fix such a $j$.\nThen, by the injection that consists in gluing a $2j$-gon on the first boundary as on Figure~\\ref{fig_reducing_boundary}, we have \n\\[\\beta_g^{(p_1+1,p_2)}(\\mathbf{f}) \\leq \\beta_g^{(p_1,p_2)}(\\mathbf{f}+\\mathbf{1}_j) \\leq C \\beta_g^{(p_1,p_2)}(\\mathbf{f}),\\]\nwhere the last inequality uses the first item of the Corollary. This proves the second point.\n\\begin{figure}\n\\center\n\\includegraphics[scale=0.8]{reducing_boundary}\n\\caption{Reducing the size of a boundary by $2$ by adding a $2j$-gon (here, the boundary is in blue, and $j=3$).}\\label{fig_reducing_boundary}\n\\end{figure}\n\\end{proof}\n\n\\paragraph{Outline of the proof of Lemma \\ref{lem_BRL}.}\nThe general idea is the same as in \\cite{BL19}, namely building an injection that removes a small piece of a map (here, we would like to remove a face of degree $2j_0$). Just like in \\cite{BL19}, this implies to merge vertices, so we will try to bound the degrees of the vertices involved, so that the number of ways to do the surgery backwards is not too high. However, since we work in a more general setting, several new constraints appear. First, the degrees of the faces are not bounded, so we must make sure that our surgery operations do not involve faces of huge degrees. This is the purpose of finding \"very nice edges\" in Section \\ref{subsec_good_sets} below. Also, we will not always be able to remove a face of degree exactly $2j_0$. We will therefore remove either a face with degree higher than $2j_0$, or several faces which combined are \"larger\" than a face of degree $2j_0$. We will then use the two (easy) Lemmas~\\ref{lem_grande_face} and~\\ref{lem_petites_faces} to conclude.\n\n\\begin{lem}\\label{lem_grande_face}\nIf $p\\geq j_0 \\geq 1$, then\n\\[ |\\mathbf{f}| \\beta_g \\left( \\mathbf{f}-\\mathbf{1}_{j_0} \\right) \\geq j_0 f_{j_0} \\, \\beta_g \\left( \\mathbf{f}-\\mathbf{1}_p \\right).\\]\nIn particular, if $j_0 f_{j_0} \\geq \\delta |\\mathbf{f}|$, then\n\\begin{equation}\\label{eq_transfer_grande_face}\n\\beta_g(\\mathbf{f}-\\mathbf{1}_{j_0}) \\geq \\delta \\beta_g(\\mathbf{f}-\\mathbf{1}_p)\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nThe second point is immediate from the first. For the first point, the right-hand side counts maps in $\\mathcal{B}_g \\left( \\mathbf{f}-\\mathbf{1}_p \\right)$ with a marked edge such that the face on its right has degree $2j_0$. The left-hand side counts maps in $\\mathcal{B}_g \\left( \\mathbf{f}-\\mathbf{1}_{j_0} \\right)$ with a marked edge, so it is enough to build an injection from the first set to the second. Take a map $m$ in $\\mathcal{B}_g \\left( \\mathbf{f}-\\mathbf{1}_p \\right)$ and mark an edge $e$ of $m$ with a face of degree $2j_0$ on its right. We glue a path of $p-j_0$ edges to the starting point of $e$, just on the right of $e$ as on Figure~\\ref{fig_plus_grande_face}. One obtains a map of $\\mathcal{B}_g \\left( \\mathbf{f}-\\mathbf{1}_{j_0} \\right)$ with a marked edge, and going backwards is straightforward.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=0.7]{plus_grande_face}\n\\caption{The injection of Lemma~\\ref{lem_grande_face} (here with $j_0=3$ and $p=5$). The marked edge is in red.}\\label{fig_plus_grande_face}\n\\end{figure}\n\\end{proof}\n\n\\begin{lem}\\label{lem_petites_faces}\nIf $1j_0$, transform the face of degree $2j_0$ into a face of degree $2d$ by adding a path of $d-j_0$ edges like in the previous proof. Then tessellate this face of degree $2d$ as on Figure~\\ref{fig_plus_petites_faces}. We obtain a map of $\\mathcal{B}_g \\left( \\mathbf{f}-\\mathbf{1}_{j_0} \\right)$ with a marked edge, and this operation is also injective.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=0.8]{plus_petites_faces}\n\\caption{The injection of Lemma~\\ref{lem_petites_faces} (here with $j_0=5$ and $(d_1,d_2,d_3)=(2,3,3)$).}\\label{fig_plus_petites_faces}\n\\end{figure}\n\\end{proof}\n\n\\subsection{Good sets of edges}\n\\label{subsec_good_sets}\n\nThe injection we will build to prove the Bounded ratio Lemma takes as input a pair $(m, E)$, where $m$ is a map and $E$ is a set of edges of $m$ satisfying the properties we will need to perform some surgery around $E$. We will call such a set a \\emph{good set}. Our goal in this subsection is to define a good set and to prove that any map contains a linear number of good sets (Proposition~\\ref{prop_good_sets}). We recall that we consider that the edges are oriented from white to black, and therefore it makes sense to define the left or right side of an edge.\n\nThroughout this section, we work under the assumptions of Lemma~\\ref{lem_BRL}. Let $m \\in \\mathcal{B}_g(\\mathbf{f})$. Let $A_1:=2A \\left( \\min \\left( \\frac{\\kappa}{32},\\delta \\right) \\right)$.\n\n\\begin{rem}\\label{rem_causal_graph}\nWe will have several different constants (depending on $A(\\cdot)$, $\\delta$ and $\\kappa$) defined in terms of each other in this subsection. To help convince the reader there is no circular dependency between them, we provide a \"causal graph\" of all the involved constants.\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=0.8]{causal_graph}\n\\end{figure}\n\\end{rem}\n\nWe say that an edge $e$ of $m$ is \\emph{nice} if it is not incident to a face of degree larger than $2A_1$.\n\\begin{fact} At least $\\left( 1-\\frac{\\kappa}{16} \\right) |\\mathbf{f}|$ of the edges in $m$ are nice.\n\\end{fact}\n\\begin{proof}\nDraw an edge $e$ of $m$ uniformly at random. The face $f$ sitting to the right of $e$ is drawn at random with a probability proportional to its degree. By the second assumption of \\eqref{eq_petites_faces}, the probability that $f$ has degree larger than $2A_1$ is less than $\\frac{\\kappa}{32}$. The same is true for the face sitting to the left of $e$.\n\\end{proof}\n\nWe will need to bound the degrees not only of the faces incident to an edge, but also of the faces close to this edge for the dual distance. More precisely, we will define the \\emph{dual distance between two edges $e_1, e_2$ of $m$} as the dual distance between the face on the right of $e_1$ and the face on the right of $e_2$. We fix a value $r$ (depending on $A_1$ and $\\kappa$) that we will specify later. Let $A_r$ be the function given by Lemma~\\ref{lem_tight_degree_in_ball} for $A(\\cdot)$ and $r$, and let $A_2=A_r \\left( \\frac{\\kappa}{16} \\right)$. We will call an edge $e$ of $m$ \\emph{very nice} if it is nice and no edge at dual distance $r$ or less from $e$ is incident to a face of degree larger than $2A_2$. By applying Lemma~\\ref{lem_tight_degree_in_ball} to the stationary random graph obtained by rooting the dual map $m^*$ at a uniform edge, the proportion of edges of $m$ at dual distance $r$ or less from a face larger than $2A_2$ is at most $\\frac{\\kappa}{16}$. Hence, we get the following observation.\n\n\\begin{fact}At least $\\left( 1-\\frac{\\kappa}{8} \\right) |\\mathbf{f}|$ of the edges of $m$ are very nice.\n\\end{fact}\n\nLet $D=\\frac{4}{\\kappa}$. By the first assumption of \\eqref{eq_petites_faces}, we know that $D$ is larger than twice the average vertex degree in $m$. Since at most half of the vertices have degree at least twice the average degree, and since there are more than $\\kappa |\\mathbf{f}|$ vertices, we have the following.\n\n\\begin{fact}\nThere are at least $\\frac{\\kappa}{4}|\\mathbf{f}|$ vertices of the same colour with degree less than $D$ in $m$.\n\\end{fact}\n\nWithout loss of generality, assume that this colour is white (we recall that the vertices are coloured black and white so that each edge joins two vertices of different colours and the root is oriented from white to black). We say that a white vertex is \\emph{fine} if it has degree at most $D$, and that an edge is \\emph{fine} if it is incident to a fine white vertex and the face on its right is not of degree $2$. By the previous fact, and since every vertex is incident to at least a face of degree $>2$, there are at least $\\frac{\\kappa}{4} |\\mathbf{f}|$ fine edges in $m$, incident to $\\frac{\\kappa}{4}|\\mathbf{f}|$ distinct white vertices.\nAn edge is said to be \\textit{good} if it is both very nice and fine. Summing up the last results, we have the following.\n\n\\begin{lem}\\label{lem_good_edges}\nThere are at least $\\frac{\\kappa }{8} |\\mathbf{f}|$ good edges in $m$, incident to $\\frac{\\kappa}{8} |\\mathbf{f}|$ distinct white vertices.\n\\end{lem}\n\nWe now fix the value of $r$ at $r=\\frac{16A_1}{\\kappa}+1$ (which is possible since $A_1$ does not depend on $r$, see Remark~\\ref{rem_causal_graph} above). We call a set $S$ of $A_1$ edges of $m$ a \\textit{good set} if all the edges of $S$ are good, they are incident to distinct white vertices and they are all at dual distance less than $2r$ from each other. Our next goal is to find a large number of good sets in the map $m$. Note that these good sets do not need to be disjoint.\n\n\\begin{prop}\\label{prop_good_sets}\nThere are at least $\\frac{\\kappa}{16} |\\mathbf{f}|$ good sets of edges in $m$.\n\\end{prop}\n\n\\begin{proof}\nThe proof follows the argument from~\\cite{BL19}.\nLet $G$ be a set of $\\frac{\\kappa }{8} |\\mathbf{f}|$ good edges incident to distinct white vertices given by Lemma~\\ref{lem_good_edges}. In this proof, the balls $B_r^*(e)$ that we will consider will be for the dual distance. We can assume that for every $e\\in G$, the ball $B_r^*(e)$ does not contain all the edges of $m$, since otherwise the proposition is obviously true.\n\nIn that case, for all $e\\in G$, since $m^*$ is connected we must have $|B^*_r(e)|>r$.\nWe are going to find a collection of distinct good sets $(S_i)$. For this, we build by induction a decreasing sequence of sets of good edges $(G_i)$, such that for each $i$, the set $G_{i+1}$ is obtained from $G_i$ by removing one element. We set $G_0=G$. Let $0 \\leq i<\\frac{\\kappa }{16} |\\mathbf{f}|$, and assume that we have built $G_0, G_1, \\dots, G_i$. Then $|G_i|=|G|-i$, so\n\\begin{align*}\n\\sum_{e\\in G_i} |B_r^*(e)|> \\left( |G|-i \\right)r >\\frac{\\kappa }{16} |\\mathbf{f}| r> A_1 |\\mathbf{f}|\n\\end{align*}\nby our choice of $r$. Therefore, there must be an \"$A_1$-overlap\", i.e. there exist $A_1$ edges whose balls of radius $r$ have a nonempty intersection. Thus they are all at distance at most $2r$ of each other, and we just found a good set $S_{i+1}$. Choose $e_{i+1} \\in S_{i+1}$ arbitrarily, and let $G_{i+1}=G_{i}\\setminus \\{e_{i+1}\\}$. This way we can build $G_i$ and $S_i$ for $1 \\leq i < \\frac{\\kappa}{16} |\\mathbf{f}|$, which proves the lemma.\n\\end{proof}\n\n\\subsection{Proof of the Bounded ratio Lemma: the injection}\n\\label{subsec_proof_BRL}\n\nWe now prove the Bounded ratio Lemma (Lemma~\\ref{lem_BRL}). We start with the easy case $j_0=1$: a marked digon can be contracted into a marked edge (see Figure~\\ref{fig_digon}), and if $f_1>\\delta |\\mathbf{f}|$, we have\n\\[|\\mathbf{f}|\\beta_g(\\mathbf{f}-\\mathbf{1}_1)>\\delta |\\mathbf{f}| \\beta_g(\\mathbf{f})\\]\nwhich yields the result. \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.5]{digon}\n\\caption{Contraction of a digon.}\\label{fig_digon}\n\\end{figure}\n\nWe can now assume that $j_0>1$ and $j_0 f_{j_0} > \\delta |\\mathbf{f}|$. The injection we will build takes as input a map of $\\mathcal{B}_g(\\mathbf{f})$ with a marked good set of edges, and outputs a map of $\\mathcal{B}_g(\\mathbf{\\tilde{f}})$ with a marked edge and some finite information (i.e. with values in a finite set whose size depends only on $\\delta, \\kappa$ and the function $A$), with $\\mathbf{\\tilde{f}}$ of the form:\n\\begin{enumerate}\n\\item[(i)]\neither\n\\begin{equation}\\label{brl_ftilde_first_condition}\n\\mathbf{\\tilde{f}}=\\mathbf{f}-\\mathbf{1}_p \\mbox{ where } j_0 \\leq p < A_1,\n\\end{equation}\n\\item[(ii)]\nor\n\\begin{equation}\\label{brl_ftilde_second_condition}\n\\mathbf{\\tilde{f}}=\\mathbf{f}-\\sum_{i=1}^k\\mathbf{1}_{d_i} \\mbox{ where } k \\leq A_1 \\mbox{ and } 1j_0-1$. Therefore, we can consider the first index $k$ such that $\\sum_{i=1}^k (d_i-1)>j_0-1$. Then we have $d_k < j_0$ by assumption, so $\\sum_{i=1}^k (d_i-1) < 2j_0-1$. Finally, by definition of $A_1$, we have $A_1 \\geq 2A(\\delta) \\geq 2j_0$ (the second inequality follows from the assumption $j_0 f_{j_0} > \\delta|\\mathbf{f}|$), so~\\eqref{eqn_brl_case_2} is indeed satisfied for this $k$.\n\nWe now choose the anchor of $S$ arbitrarily, and apply Step 1. We then apply Step 2 with $d=1+\\sum_{i=1}^k (d_i-1)$, i.e. we delete $d$ good vertices, including the ones incident to $e_1, \\ldots, e_k$. Note that at this point, all the faces that were incident to the edges $e_1, \\dots, e_k$ have been destroyed. We then apply Step 3 and obtain a map $m_3$ with face degree sequence $\\mathbf{f'}$. The first two assumptions on $\\mathbf{f'}$ in Step 4 are satisfied for the same reason as in Case 1, with $\\mathbf{f'}$ of the form $\\mathbf{f}+\\mathbf{1}_F-\\mathbf{1}_{j_1}-\\ldots-\\mathbf{1}_{j_{\\ell}}$. Moreover, since the faces incident to $e_1, \\dots, e_k$ have been destroyed previously, up to reordering the $j_i$'s, we may assume $j_i=d_i$ for $1 \\leq i \\leq k$. Therefore, by our choice of $d$, the third assumption~\\eqref{eq_condition_BRL} of Step 4 is also satisfied. After applying Step 4, we obtain a map $m_4$ with face degree sequence\n\\[\\mathbf{\\tilde{f}}=\\mathbf{f}-\\sum_{i=1}^{k} d_i.\\]\nFinally, we have $10$. Indeed, this is true for the root face since the root face of $M$ has degree $j$ with probability $j \\alpha_j$ for all $j \\geq 1$, and this can be extended to all faces using stationarity with respect to the simple random walk on the dual of $M$. Therefore, if $m$ has a face of degree $2j$ with $\\alpha_j=0$, then $\\P \\left( m \\subset M \\right)=0$.\n\nIf not, let $\\mathbf{h}^{(0)}$ be the internal face degree sequence of $m$. Then $f^n_j \\to +\\infty$ as $n \\to +\\infty$ for every $j$ such that $h^{(0)}_j>0$, so $\\mathbf{h}^{(0)} \\leq\\mathbf{f}^{n}$ for $n$ large enough. In particular, we are in position to use Lemma~\\ref{lem_calcul_planar}. The proof is now exactly the same as in~\\cite{BL19}: we use the fact that $\\P \\left( m \\subset M \\right)$ can be expressed using the number of ways to fill the holes of $m$ with maps of multipolygons, which is given by the left-hand side of Lemma~\\ref{lem_calcul_planar}.\n\\end{proof}\n\nWe now move on to one-endedness. The proof is quite similar, and relies on the following estimate.\n\n\\begin{lem}\\label{lem_calcul_OE}\nWe fix $(\\mathbf{f}^{n})_{n \\geq 1}$ and $(g_n)_{n \\geq 1}$ which satisfy the assumptions of Theorem~\\ref{thm_main_more_general}. We also fix $\\mathbf{h}^{(0)}$ a face degree sequence such that $\\mathbf{h}^{(0)} \\leq\\mathbf{f}^{n}$ for $n$ large enough.\n\\begin{itemize}\n\\item\nLet $k\\geq 1$, numbers $\\ell_1, \\ell_2,\\ldots,\\ell_k$ \\textbf{not all equal to $1$} and perimeters $p_i^j$ for $1\\leq j\\leq k$ and $1\\leq i\\leq \\ell_j$. Then\n\\[\\sum_{\\substack{\\mathbf{h}^{(1)}+\\mathbf{h}^{(2)}+\\ldots+\\mathbf{h}^{(k)}=\\mathbf{f}^n-\\mathbf{h}^{(0)} \\\\ g^{(1)}+g^{(2)}+\\ldots+g^{(k)}=g_n-\\sum_j(\\ell_j-1)}}\\prod_{j=1}^k \\beta_{g^{(j)}}^{(p^j_1,p^j_2,\\ldots,p^j_{\\ell_j})}(\\mathbf{h}^{(j)})=o\\left(\\beta_{g_n}(\\mathbf{f}^n)\\right)\\]\nas $n\\rightarrow \\infty$.\n\\item \nLet $k\\geq 1$ and perimeters $p_1,\\ldots,p_k$. There is a constant $C$ (that may depend on everything above) such that for every $a$ and $n$ large enough we have\n\\[\\sum_{\\substack{\\mathbf{h}^{(1)}+\\mathbf{h}^{(2)}+\\ldots+\\mathbf{h}^{(k)}=\\mathbf{f}^n-\\mathbf{h}^{(0)} \\\\ g^{(1)}+g^{(2)}+\\ldots+g^{(k)}=g_n\\\\ |\\mathbf{h}^{(1)}|, |\\mathbf{h}^{(2)}|>a}} \\, \\prod_{j=1}^k \\beta_{g_j}^{(p_j)}(\\mathbf{h}^{(j)})\\leq \\frac{C}{a} \\beta_{g_n}(\\mathbf{f}^n).\\]\n\\end{itemize}\n\\end{lem}\n\n\\begin{corr}\\label{cor_OE}\nLet $(\\mathbf{f}^{n})_{n \\geq 1}$ and $(g_n)_{n \\geq 1}$ satisfy the assumptions of Theorem~\\ref{thm_main_more_general}. Every subsequential limit $M$ of $\\left( M_{\\mathbf{f}^n}, g_n \\right)$ for $d^*_{\\mathrm{loc}}$ is a.s. one-ended in the sense that, for every finite map $m$ with holes such that $m \\subset M$, only one hole of $m$ is filled with infinitely many faces\\footnote{This is a \"weak\" definition of one-endedness. For example, it does not prevent $m$ to be the dual of a tree. However, once we will have proved that $M$ has finite vertex degrees, this will be equivalent to the usual definition.}.\n\\end{corr}\n\n\\begin{proof}\nThe proof given Lemma~\\ref{lem_calcul_OE} is exactly the same as~\\cite[Corollary 9]{BL19}, except for the additional assumption in Lemma~\\ref{lem_calcul_OE} that $\\mathbf{h}^{(0)} \\leq\\mathbf{f}^{n}$ for $n$ large enough. We take care of it in the same way as in the proof of Corollary~\\ref{cor_planar}.\n\\end{proof}\n\n\\subsection{Finiteness of the root degree}\n\\label{subsec_finite_degrees}\n\nWe now finish the proof of tightness for $d_{\\mathrm{loc}}$ (Proposition~\\ref{prop_tightness_dloc_univ}). Let $M$ be a subsequential limit of $\\left( M_{\\mathbf{f}^n, g_n} \\right)$ for $d_{\\mathrm{loc}}^*$. By Lemma~\\ref{lem_dual_convergence_univ}, to get tightness for $d_{\\mathrm{loc}}$, we need to show that almost surely, all the vertices of $M$ have finite degree. Our argument is now very similar to \\cite{BL19} and inspired by \\cite{AS03}: we will first study the degree of the root vertex by using the Bounded ratio Lemma, and then extend finiteness by using invariance under the simple random walk.\n\n\\begin{lem}\\label{lem_root_degree_is_finite_univ}\nThe root vertex of $M$ has a.s. finite degree.\n\\end{lem}\n\n\\begin{proof}\nFollowing the approach of \\cite{AS03}, we perform a filled-in lazy peeling exploration of $M$. Note that we already know by Corollary~\\ref{cor_planar} that the explored part will always be planar, so no peeling step will merge two different existing holes. Moreover, by Corollary~\\ref{cor_OE}, if a peeling step separates the boundary into two holes, then one of them is finite and will be filled with a finite map. Therefore, at each step, the explored part will have only one hole.\n\nThe peeling algorithm $\\mathcal{A}$ that we use is the following: if the root vertex $\\rho$ belongs to $\\partial m$, then $\\mathcal{A}(m)$ is the edge on $\\partial m$ on the left of $\\rho$. If $\\rho \\notin \\partial m$, then the exploration is stopped. Let $\\tau$ be the time at which the exploration is stopped. Since only finitely many edges incident to $\\rho$ are added at each step, it is enough to prove $\\tau<+\\infty$ a.s.. We recall that $\\mathcal{E}_t^{\\mathcal{A}}(M)$ is the explored part at time $t$.\n\nWe will prove that at each step, conditionally on $\\mathcal{E}_t^{\\mathcal{A}}(M)$, the probability to swallow the root and finish the exploration in a bounded amount of time is bounded from below by a positive constant. We fix $j^* \\geq 2$ with $\\alpha_{j^*}>0$. Note that such a $j^*$ exists because of the assumption $\\theta<\\frac{1}{2} \\sum_{j \\geq 1} (j-1) \\alpha_j$. For every map $m$ with one hole such that $\\rho \\in \\partial m$, we denote by $m^+$ the map constructed from $m$ as follows (see Figure~\\ref{fig_swallowing_root_univ}):\n\\begin{itemize}\n\\item\nwe first glue a \"face\" of degree $2j^*$ to $m$ along the edge of $\\partial m$ on the left of $\\rho$;\n\\item\nwe then glue together the two edges of the boundary incident to $\\rho$ together;\n\\item\nduring the next $j^*-2$ steps, at each step, we pick two consecutive edges of the boundary according to some fixed convention and glue them together.\n\\end{itemize}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.7]{swallow_root}\n\\caption{The construction of $m^+$ from $m$. In gray, the map $m$. In red, the root vertex. In blue, the new face. Here, $|\\partial m|=j^*=3$.}\n\\label{fig_swallowing_root_univ}\n\\end{figure}\n\nNote that $m^+$ is a planar map with the same perimeter as $m$ but one more face (of degree $2j^*$). By the choice of our peeling algorithm, if we have $\\tau \\geq t$ and $\\mathcal{E}_t^{\\mathcal{A}}(M)^+ \\subset M$, then we have $\\tau \\leq t+2$. Hence it is enough to prove that the quantity\n\\[ \\P \\left( m^+ \\subset M | m \\subset M \\right)\\]\nis bounded from below over finite, planar maps $m$ with one hole such that $\\rho \\in \\partial m$.\n\nWe fix such an $m$, with half-perimeter $p$ and internal face degrees given by $\\mathbf{h}$. Along some subsequence, we have $M_{\\mathbf{f}^{n},g_{n}} \\to M$ in distribution (for $d_{\\mathrm{loc}}^*$). Along the same subsequence, it holds that\n\\[ \\P \\left( m^+ \\subset M | m \\subset M \\right)\\hspace{-0.1cm}=\\hspace{-0.1cm}\\lim_{n \\to +\\infty}\\hspace{-0.1cm} \\frac{\\P \\left( m^+ \\in M_{\\mathbf{f}^{n},g_{n}} \\right)}{\\P \\left( m\\in M_{\\mathbf{f}^{n},g_{n}} \\right)} \\hspace{-0.1cm} = \\hspace{-0.1cm}\\lim_{n \\to +\\infty}\\hspace{-0.1cm} \\frac{\\beta^{(p)}_{g_n} \\left( \\mathbf{f}^n-\\mathbf{h}-\\mathbf{1}_{j^*} \\right)}{\\beta^{(p)}_{g_n} \\left( \\mathbf{f}^n-\\mathbf{h} \\right)}. \\]\nBy our choice of $j^*$, we have $f^n_{j^*} \\geq \\frac{\\alpha_{j^*}}{2} |\\mathbf{f}^n|$ for $n$ large enough, so we can apply the Bounded ratio Lemma, which concludes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_tightness_dloc_univ}]\nLet $M$ be a subsequential limit of $\\left( M_{\\mathbf{f}^{n},g_{n}} \\right)$. We recall that for all $n$, the map $\\left( M_{\\mathbf{f}^{n},g_{n}} \\right)$ is stationary for the simple random walk on its vertices. Therefore, by Lemma~\\ref{lem_root_degree_is_finite_univ} and the same argument as in \\cite{AS03} (see also the proof of Lemma~\\ref{lem_easy_dual_convergence} above), almost surely all the vertices of $M$ have finite degree. By Lemma~\\ref{lem_dual_convergence_univ}, this guarantees that $\\left( M_{\\mathbf{f}^{n},g_{n}} \\right)$ is tight for $d_{\\mathrm{loc}}$.\n\nThe a.s. planarity of $M$ is proved in~Corollary \\ref{cor_planar}. Finally, it easy to check that for maps with finite vertex degrees, the weak version of one-endedness proved in Corollary~\\ref{cor_OE} implies the usual one. Indeed, if $V$ is a finite set of vertices of $M$, one can consider a finite, connected submap of $M$ containing all the faces and edges incident to vertices of $V$. Then Corollary~\\ref{cor_planar} ensures that this submap does not separate $M$ into two infinite maps.\n\\end{proof}\n\n\\section{Weakly Markovian bipartite maps}\\label{sec_univ_markov}\n\nOur goal in this Section is to prove Theorem~\\ref{thm_weak_Markov_general}.\n\n\\paragraph{Weakly Markovian bipartite maps.}\nFor a finite, bipartite map $m$ with one hole, we denote by $|\\partial m|$ the half-perimeter of the hole of $m$. For all $j \\geq 1$, we also denote by $v_j(m)$ the number of internal faces of $m$ with degree $2j$.\n\n\\begin{defn}\\label{defn_weak_Markov}\nLet $M$ be a random infinite, one-ended, bipartite planar map. We say that $M$ is \\emph{weakly Markovian} if for every finite map $m$ with one hole, the probability $\\P \\left( m \\subset M \\right)$ only depends on $|\\partial m|$ and $\\left( v_j(m) \\right)_{j \\geq 1}$.\n\\end{defn}\n\nLet $\\mathcal{V}$ be the set of sequences $\\mathbf{v}=(v_j)_{j \\geq 1}$ such that $v_j=0$ for $j$ large enough. If $M$ is weakly Markovian and $\\mathbf{v} \\in \\mathcal{V}$, we will denote by $a^p_{\\mathbf{v}}$ the probability $\\P \\left( m \\subset M \\right)$ for a map $m$ with $|\\partial m|=p$ and $v_j(m)=v_j$ for all $j$. Note that this only makes sense if there is such a map $m$, which is equivalent to\n\\begin{equation}\\label{eqn_good_pvv}\np \\leq 1+\\sum_{j \\geq 1} (j-1)v_j.\n\\end{equation}\nTherefore, if $p \\geq 1$, we will denote by $\\mathcal{V}_p \\subset \\mathcal{V}$ the set of those $\\mathbf{v}$ satisfying \\eqref{eqn_good_pvv}. Note that $\\mathcal{V}_1=\\mathcal{V}$.\nIn particular, by definition, for $\\mathbf{q} \\in \\mathcal{Q}_h$, the $\\mathbf{q}$-IBPM is weakly Markovian, and the corresponding constants $a^p_{\\mathbf{v}}$ are:\n\\[a^p_{\\mathbf{v}}(\\mathbf{q}):= C_p(\\mathbf{q}) \\mathbf{q}^{\\mathbf{v}},\\]\nwhere $\\mathbf{q}^{\\mathbf{v}} := \\prod_{j \\geq 1} q_j^{v_j}$. Therefore, if $M$ is of the form $\\mathbb M_{\\mathbf{Q}}$ for some random weight sequence $\\mathbf{Q}$, we have $a^p_{\\mathbf{v}} = \\mathbb E[ C_p(\\mathbf{Q}) \\mathbf{Q}^{\\mathbf{v}} ]$.\n\n\\paragraph{Sketch of the proof of Theorem~\\ref{thm_weak_Markov_general}.}\nWe first note that the second point of Theorem~\\ref{thm_weak_Markov_general} is immediate once the first point is proved. Indeed, let us write $\\mathrm{Rootface}(m)$ for the degree of the root face of a map $m$. If $\\mathrm{Rootface}(M)$ has finite expectation, then\n\\[ \\mathbb E \\left[ \\mathbb E \\left[ \\mathrm{Rootface}(\\mathbb M_{\\mathbf{Q}}) | \\mathbf{Q} \\right] \\right] = \\mathbb E \\left[ \\mathrm{Rootface}(\\mathbb M_{\\mathbf{Q}}) \\right] <+\\infty, \\]\nso $\\mathbb E \\left[ \\mathrm{Rootface}(\\mathbb M_{\\mathbf{Q}}) | \\mathbf{Q} \\right]<+\\infty$ a.s., so $\\mathbf{Q} \\in \\mathcal{Q}_f$ a.s..\n\nThe first point of Theorem~\\ref{thm_weak_Markov_general} is the natural analogue of Theorem~2 of \\cite{BL19}, where triangulations are replaced by more general maps. The proof will rely on similar ideas: we fix a weakly Markovian map $M$ with associated constants $a^p_{\\mathbf{v}}$, and we would like to find a random $\\mathbf{Q}$ such that $a^p_{\\mathbf{v}} = \\mathbb E[ C_p(\\mathbf{Q}) \\mathbf{Q}^{\\mathbf{v}} ]$ for all $p$ and $\\mathbf{v}$. We will use peeling equations to establish inequalities between the $a^p_{\\mathbf{v}}$, and the existence of $\\mathbf{Q}$ will follow from the Hausdorff moment problem. However, compared to~\\cite{BL19}, two new difficulties arise:\n\\begin{itemize}\n\\item[$\\bullet$]\nthe random weights $\\mathbf{q}$ form a family of real numbers instead of just one real number;\n\\item[$\\bullet$]\nin the triangular case, with the notations of Definition~\\ref{defn_weak_Markov}, it was immediate that all the numbers $a^p_{\\mathbf{v}}$ are determined by the numbers $a^1_{\\mathbf{v}}$. This is not true anymore.\n\\end{itemize}\nThe first issue can be handled by using the multi-dimensional version of the Hausdorff moment problem. The second one, on the other hand, will make the proof a bit longer than in \\cite{BL19}. More precisely, the Hausdorff moment problem will now provide us, for every $p \\geq 1$, a $\\sigma$-finite measure $\\mu_p$ on the set of weight sequences, which describes $a^p_{\\mathbf{v}}$ for all $\\mathbf{v}$. We will then use the peeling equations to prove that all the $\\mu_p$ are actually determined by $\\mu_1$.\n\nBecause of the condition \\eqref{eqn_good_pvv}, we will need to find a measure with suitable $\\mathbf{v}$-th moments for all $\\mathbf{v} \\in \\mathcal{V}_p$, which is slightly different than the usual Hausdorff moment problem where $\\mathbf{v} \\in \\mathcal{V}$. Therefore, we first need to state a suitable version of the moment problem, which will follow from the usual one. This is done in the next subsection.\n\n\\subsection{The incomplete Hausdorff moment problem}\n\nTo state our version of the moment problem (Proposition~\\ref{prop_moment_Hausdorff} below), we will need to consider the space of sequences $\\left( u_{\\mathbf{v}} \\right)_{\\mathbf{v} \\in \\mathcal{V}_p}$. For $j \\geq 1$, we denote by $\\Delta_j$ the discrete derivation operator on the $j$-th coordinate on this space. That is, if $u=(u_{\\mathbf{v}})$, we write\n\\[ \\left( \\Delta_j u \\right)_{\\mathbf{v}} = u_{\\mathbf{v}}-u_{\\mathbf{v}+\\mathbf{1}_j}.\\]\nIt is easy to check that the operators $\\Delta_j$ commute with each other. For all $\\mathbf{k}=(k_j)_{j \\geq 1}$ such that $k_j=0$ for $j$ large enough (say for $j \\geq j_0$), we define the operator $\\Delta^{\\mathbf{k}}$ by\n\\[\n\\Delta^{\\mathbf{k}} u= \\Delta_1^{k_1} \\Delta_2^{k_2} \\dots \\Delta_{j_0}^{k_{j_0}} u.\n\\]\nIn other words, we have\n\\[\\Delta^{\\mathbf{k}} u= \\sum_{\\mathbf{i}} \\left( \\prod_{j \\geq 1} (-1)^{i_j} \\binom{k_j}{i_j} \\right) u_{\\mathbf{v}+\\mathbf{i}}, \\]\nwhere the sum is over families $\\mathbf{i}=(i_j)_{j \\geq 1}$, and the terms with a nonzero contribution are those for which $0 \\leq i_j \\leq k_j$ for every $j \\geq 1$. The \"usual\" Hausdorff moment problem is then the following.\n\n\\begin{thm}\\label{thm_hausdorff_usual}\nLet $(u_{\\mathbf{v}})_{\\mathbf{v} \\in \\mathcal{V}}$ be such that, for any $\\mathbf{v} \\in \\mathcal{V}$ and any $\\mathbf{k} \\geq \\mathbf{0}$, we have\n\\[ \\Delta^{\\mathbf{k}} u_{\\mathbf{v}} \\geq 0.\\]\nThen there is a unique measure $\\mu$ on $\\mathcal{Q}=[0,1]^{\\mathbb N^*}$ (equipped with the product $\\sigma$-algebra) such that, for all $\\mathbf{v} \\in \\mathcal{V}$, we have\n\\[ u_{\\mathbf{v}}=\\int \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q). \\]\nIn particular $\\mu$ is finite, with total mass $u_{\\mathbf{0}}$.\n\\end{thm}\nMore precisely, this is the infinite-dimensional Hausdorff moment problem, which can be deduced immediately from the finite-dimensional one by the Kolmogorov extension theorem.\n\nFor $p \\geq 1$, we recall that $\\mathcal{V}_p \\subset \\mathcal{V}$ is the set of $\\mathbf{v} \\in \\mathcal{V}$ that satisfy $\\sum_{j \\geq 1} (j-1) v_j \\geq p-1$. We also denote by $\\mathcal{V}_p^*$ the set of $\\mathbf{v} \\in \\mathcal{V}_p$ for which there is $j \\geq 2$ such that $v_j>0$ and $\\mathbf{v}-\\mathbf{1}_j \\in \\mathcal{V}_p$. In other words $\\mathcal{V}_p^*$ can be thought of as the \"interior\" of $\\mathcal{V}_p$. Finally, we recall that\n\\[ \\mathcal{Q}^*=\\{ \\mathbf{q} \\in [0,1]^{\\mathbb N^*} | \\exists j \\geq 2, q_j>0\\}.\\]\n\n\\begin{prop}\\label{prop_moment_Hausdorff}\nFix $p \\geq 1$, and let $\\left( u_{\\mathbf{v}} \\right)_{\\mathbf{v} \\in \\mathcal{V}_p}$ be a family of real numbers. We assume that for all $\\mathbf{v} \\in \\mathcal{V}_p$ and all $\\mathbf{k} \\geq \\mathbf{0}$, we have\n\\[ \\Delta^{\\mathbf{k}} u_{\\mathbf{v}} \\geq 0.\\]\nThen there is a $\\sigma$-finite measure $\\mu$ on $\\mathcal{Q}^*$ such that, for all $\\mathbf{v} \\in \\mathcal{V}_p^*$, we have\n\\[ u_{\\mathbf{v}}=\\int \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q). \\]\nMoreover, if $p=1$, then $\\mu$ is finite and $\\mu(\\mathcal{Q}^*) \\leq u_\\mathbf{0}$.\n\\end{prop}\n\nNote that this version is \"weaker\" than Theorem~\\ref{thm_hausdorff_usual} in the sense that it is not always possible to have $u_{\\mathbf{v}}=\\int \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q)$ for $\\mathbf{v} \\in \\mathcal{V}_p \\backslash \\mathcal{V}^*_p$. A simple example of this phenomenon in dimension one is that the sequence $\\left( \\mathbbm{1}_{i=1} \\right)_{i \\geq 1}$ has all its discrete derivatives nonnegative. However, there is no measure on $[0,1]$ with first moment $1$ and all higher moments $0$. On the other hand, we can assume an additional property of our measure $\\mu$, namely it is supported by $\\mathcal{Q}^*$ instead of $\\mathcal{Q}$ in Theorem~\\ref{thm_hausdorff_usual}.\n\n\\begin{proof}\nWe start with the case $p=1$. Then $\\mathcal{V}_1=\\mathcal{V}$, so by Theorem~\\ref{thm_hausdorff_usual}, there is a measure $\\widetilde{\\mu}$ on $\\mathcal{Q}$ such that, for all $\\mathbf{v} \\in \\mathcal{V}_1$, we have\n\\[u_{\\mathbf{v}} = \\int_{\\mathcal{Q}} \\mathbf{q}^{\\mathbf{v}} \\widetilde{\\mu}(\\mathrm{d} \\q). \\]\nLet $\\mu$ be the restriction of $\\widetilde{\\mu}$ to $\\mathcal{Q}^*$. If $\\mathbf{v} \\in \\mathcal{V}_1^*$ and $\\mathbf{q} \\in \\mathcal{Q} \\backslash \\mathcal{Q}^*$, then there is $j \\geq 2$ such that $\\mathbf{v}_j >0$ but $q_j=0$, so $\\mathbf{q}^{\\mathbf{v}}=0$. It follows that, for all $\\mathbf{v} \\in \\mathcal{V}_1^*$, we have\n\\[ \\int_{\\mathcal{Q}^*} \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q) = \\int_{\\mathcal{Q}} \\mathbf{q}^{\\mathbf{v}} \\widetilde{\\mu}(\\mathrm{d} \\q) = u_{\\mathbf{v}}.\\]\nMoreover, the total mass of $\\mu$ is not larger than the total mass of $\\widetilde{\\mu}$, so it is at most $u_{\\mathbf{0}}$.\n\nWe now assume $p \\geq 2$. Let $\\mathbf{v} \\in \\mathcal{V}_p$. Then $\\mathbf{v}+\\mathbf{w} \\in \\mathcal{V}_p$ for all $\\mathbf{w} \\in \\mathcal{V}$, so the sequence $\\left( u_{\\mathbf{v}+\\mathbf{w}} \\right)_{\\mathbf{w} \\in \\mathcal{V}}$ satisfies the assumptions of Theorem~\\ref{thm_hausdorff_usual}. Therefore, there is a finite measure $\\mu_{\\mathbf{v}}$ on $\\mathcal{Q}$ such that\n\\[ u_{\\mathbf{v}+\\mathbf{w}} = \\int \\mathbf{q}^{\\mathbf{w}} \\mu_{\\mathbf{v}}(\\mathrm{d} \\q)\\]\nfor all $\\mathbf{w} \\in \\mathcal{V}$. Now let $\\mathbf{v}, \\mathbf{v}' \\in \\mathcal{V}_p$. For all $\\mathbf{w}$, we have\n\\[ \\int \\mathbf{q}^{\\mathbf{v}'} \\mathbf{q}^{\\mathbf{w}} \\mu_{\\mathbf{v}}(\\mathrm{d} \\q)= u_{\\mathbf{v}+\\mathbf{v}'+\\mathbf{w}}=\\int \\mathbf{q}^{\\mathbf{v}} \\mathbf{q}^{\\mathbf{w}} \\mu_{\\mathbf{v}'}(\\mathrm{d} \\q).\\]\nIn other words, the measures $\\mathbf{q}^{\\mathbf{v}'} \\mu_{\\mathbf{v}}(\\mathrm{d} \\q)$ and $\\mathbf{q}^{\\mathbf{v}} \\mu_{\\mathbf{v}'}(\\mathrm{d} \\q)$ have the same moments, so by uniqueness in Theorem~\\ref{thm_hausdorff_usual}\n\\begin{equation}\\label{eqn_consistence_muv}\n\\mathbf{q}^{\\mathbf{v}'} \\mu_{\\mathbf{v}}(\\mathrm{d} \\q)=\\mathbf{q}^{\\mathbf{v}} \\mu_{\\mathbf{v}'}(\\mathrm{d} \\q).\n\\end{equation}\nIn particular, for all $\\mathbf{v} \\in \\mathcal{V}_p$, we can consider the $\\sigma$-finite measure \\[\\widetilde{\\mu}_{\\mathbf{v}}(\\mathrm{d} \\q)= \\frac{\\mu_{\\mathbf{v}}(\\mathrm{d} \\q)}{\\mathbf{q}^{\\mathbf{v}}}\\] defined on $\\{ \\mathbf{q}^{\\mathbf{v}}>0\\}$. Then \\eqref{eqn_consistence_muv} implies that, for any $\\mathbf{v}, \\mathbf{v}' \\in \\mathcal{V}_p$, the measures $\\widetilde{\\mu}_{\\mathbf{v}}$ and $\\widetilde{\\mu}_{\\mathbf{v}'}$ coincide on $\\{ \\mathbf{q}^{\\mathbf{v}}>0 \\} \\cap \\{ \\mathbf{q}^{\\mathbf{v}'}>0\\}$. Therefore, there is a measure $\\mu$ on $\\bigcup_{\\mathbf{v} \\in \\mathcal{V}_p} \\{\\mathbf{q}^{\\mathbf{v}}>0\\} = \\mathcal{Q}^*$ such that, for all $\\mathbf{v} \\in \\mathcal{V}_p$, we have\n\\begin{equation}\\label{eqn_mu_and_muv_weak}\n\\mu_{\\mathbf{v}}(\\mathrm{d} \\q) = \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q) \\quad \\mbox{ on } \\quad \\{\\mathbf{q}^{\\mathbf{v}}>0\\}.\n\\end{equation}\nSince $\\mu$ is finite on $\\{q_j>\\varepsilon\\}$ for all $\\varepsilon>0$ and $j \\geq 2$, the measure $\\mu$ is $\\sigma$-finite. We would now like to extend the equality~\\eqref{eqn_mu_and_muv_weak} to all $\\mathcal{Q}^*$ under the condition $\\mathbf{v} \\in \\mathcal{V}_p^*$.\n\nFor this, let $\\mathbf{v} \\in \\mathcal{V}_p^*$, and let $j \\geq 2$ be such that $v_j>0$ and $\\mathbf{v}-\\mathbf{1}_j \\in \\mathcal{V}_p$. We have $p \\mathbf{1}_j \\in \\mathcal{V}_p$, so we can apply \\eqref{eqn_consistence_muv} to $\\mathbf{v}$ and $p \\mathbf{1}_j$. We obtain, on $\\{ q_j>0 \\}$:\n\\[ \\mu_{\\mathbf{v}}(\\mathrm{d} \\q) = \\mathbf{q}^{\\mathbf{v}} \\frac{\\mu_{p \\mathbf{1}_j}(\\mathrm{d} \\q)}{q_j^p} = \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q), \\]\nusing also \\eqref{eqn_mu_and_muv_weak} for $p \\mathbf{1}_j$. In other words, \\eqref{eqn_mu_and_muv_weak} holds on $\\{ q_j>0 \\}$.\n\nOn the other hand, for $\\mathbf{v}$ and $\\mathbf{v}-\\mathbf{1}_p$, we can obtain a stronger version of \\eqref{eqn_consistence_muv}. More precisely, for all $\\mathbf{w}$, we have\n\\[ \\int q_j \\mathbf{q}^{\\mathbf{w}} \\mu_{\\mathbf{v}-\\mathbf{1}_j}(\\mathrm{d} \\q) = u_{\\mathbf{v}+\\mathbf{w}} = \\int \\mathbf{q}^{\\mathbf{w}} \\mu_{\\mathbf{v}}(\\mathrm{d} \\q),\\]\nso the measures $q_j \\mu_{\\mathbf{v}-\\mathbf{1}_j}(\\mathrm{d} \\q)$ and $\\mu_{\\mathbf{v}}(\\mathrm{d} \\q)$ have the same moments, so they coincide. But the first one is $0$ on $\\{q_j=0\\}$, so it is also the case for the second. Therefore, \\eqref{eqn_mu_and_muv_weak} holds on $\\{ q_j=0 \\}$, with both sides equal to $0$.\n\nTherefore, we have proved that \\eqref{eqn_mu_and_muv_weak} holds on $\\mathcal{Q}$. By integrating over $\\mathcal{Q}^*$ and using that the total mass of $\\mu_{\\mathbf{v}}$ is $u_{\\mathbf{v}}$ and is supported by $\\mathcal{Q}^*$, we get the result.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thm_weak_Markov_general}}\n\nAs in \\cite{BL19}, we start by writing down the peeling equations, which are linear equations between the numbers $a^p_{\\mathbf{v}}$ together. For every $p \\geq 1$ and $\\mathbf{v} \\in \\mathcal{V}_p$, we have\n\\begin{equation}\\label{eqn_peeling_equation_bipartite}\na^p_{\\mathbf{v}}= \\sum_{j \\geq 1} a^{p+j-1}_{\\mathbf{v}+\\mathbf{1}_j}+2\\sum_{i=1}^{p-1} \\sum_{\\mathbf{w} \\in \\mathcal{V}} \\beta_0^{(i-1)}(\\mathbf{w}) a^{p-i}_{\\mathbf{v}+\\mathbf{w}},\n\\end{equation}\nwhere we recall that $\\beta_0^{(i-1)}(\\mathbf{w})$ is the number of planar, bipartite maps of the $2(i-1)$-gon with exactly $w_j$ internal faces of degree $2j$ for all $j \\geq 1$. These equations, together with the facts that $a^1_\\mathbf{0}=1$ and $a^p_{\\mathbf{v}} \\geq 0$, characterize the families $(a^p_{\\mathbf{v}})$ of numbers that may arise from a weakly Markovian map. In order to be able to use the Hausdorff moment problem, we now need to check that the discrete derivatives of $(a^p_{\\mathbf{v}})$ are nonnegative.\n\n\\begin{lem}\\label{lem_abs_monotone}\nLet $M$ be a weakly Markovian bipartite map, and let $\\left( a^p_{\\mathbf{v}} \\right)$ be the associated constants. For every $\\mathbf{k} \\geq \\mathbf{0}$, $p \\geq 1$ and $\\mathbf{v} \\in \\mathcal{V}_p$, we have\n\\[ \\left( \\Delta^{\\mathbf{k}} a^p \\right)_{\\mathbf{v}} \\geq 0.\\]\n\\end{lem}\n\n\\begin{proof}\nThe proof is similar to the proof of Lemma 16 in \\cite{BL19}, with the following modification: in \\cite{BL19}, it was useful that in the same peeling equation, we had $a^p_v$ appearing on the left and $a^p_{v+1}$ on the right. However, in \\eqref{eqn_peeling_equation_bipartite} $a^p_{\\mathbf{v}+\\mathbf{1}_j}$ does not appear in the right-hand side (this is because we are using the lazy peeling process of \\cite{Bud15} instead of the simple peeling of \\cite{Ang03}). Therefore, instead of using directly the peeling equation, we will need to use the \\emph{double peeling equation}, which corresponds to performing two peeling steps, instead of one in~\\eqref{eqn_peeling_equation_bipartite}.\n\nMore precisely, the peeling equation \\eqref{eqn_peeling_equation_bipartite} gives an expansion of $a^p_{\\mathbf{v}}$. The \\emph{double peeling equation} is obtained from \\eqref{eqn_peeling_equation_bipartite} by replacing all the terms in the right-hand side by their expansion given by \\eqref{eqn_peeling_equation_bipartite}. Note that this indeed makes sense because if $\\mathbf{v} \\in \\mathcal{V}_p$, then $\\mathbf{v}+\\mathbf{1}_j \\in \\mathcal{V}_{p+j-1}$ for all $j \\geq 1$, and $\\mathbf{v}+\\mathbf{w} \\in \\mathcal{V}_{p-i}$ for all $i \\geq 1$ and $\\mathbf{w} \\in \\mathcal{V}$.\n\nThe equation we obtain is of the form\n\\begin{equation}\\label{eqn_double_peeling_equation}\na^p_{\\mathbf{v}}=\\sum_{i \\in \\mathbb Z, \\, \\mathbf{w} \\in \\mathcal{V}} c^{p,i}_{\\mathbf{v},\\mathbf{w}} a^{p+i}_{\\mathbf{v}+\\mathbf{w}},\n\\end{equation}\nwhere the coefficients $c^{p,i}_{\\mathbf{v},\\mathbf{w}}$ are nonnegative integers. An explicit formula for these could be computed in terms of the $\\beta_0^{(i)}(\\mathbf{w})$, but this will not be needed. Here are the facts that will be useful:\n\\begin{enumerate}\n\\item\nthe coefficients $c^{p,i}_{\\mathbf{v},\\mathbf{w}}$ actually do not depend on $\\mathbf{v}$, so we can write them $c^{p,i}_{\\mathbf{w}}$,\n\\item\\label{item_coeff_equal_one}\nwe have $c^{p,2j-2}_{2\\cdot\\mathbf{1}_j}=1$ for every $j \\geq 1$,\n\\item\\label{item_coeff_positive}\nwe have $c^{p,0}_{\\mathbf{1}_j} \\geq 1$ for every $j \\geq 2$.\n\\end{enumerate}\nThe first item follows from the fact that at each time, the available next peeling steps do not depend on the internal face degrees of the explored region.\nThe second item expresses the fact that, for a given peeling algorithm, there is a unique way to obtain a map with half-perimeter $p+2j-2$ with internal faces $\\mathbf{v}+2\\cdot\\mathbf{1}_j$ in two peeling steps. This way is to discover a unique face of degree $2j$ at both steps. The third item means that it is possible (not necessarily in a unique way) to obtain in two peeling steps a map with the same perimeter but one more face of degree $2j$. This is achieved by discovering a new face of degree $2j$ at the first step, and gluing all but two sides of this face two by two at the second step (see Figure~\\ref{figure_pconstant_vincrease}). \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.7]{one_more_face}\n\\caption{In two peeling steps, the perimeter stays constant and one face with degree $2j$ is added (here $j=3$).}\\label{figure_pconstant_vincrease}\n\\end{center}\n\\end{figure}\n\nWe now prove the lemma by induction on $|\\mathbf{k}|=\\sum_{j \\geq 1} k_j$. First, the case $|\\mathbf{k}|=0$ just means that $a^p_{\\mathbf{v}} \\geq 0$ for all $p \\geq 1$ and $\\mathbf{v} \\in \\mathcal{V}_p$, which is immediate. Let us now assume that the lemma is true for $\\mathbf{k}$ and prove it for $\\mathbf{k}+\\mathbf{1}_j$, where $j \\geq 1$. We will first treat the case where $j \\geq 2$.\nUsing the double peeling equation \\eqref{eqn_double_peeling_equation} for $(p, \\mathbf{v}+\\mathbf{i})$ for different values of $\\mathbf{i}$, we have\n\\[ \\left( \\Delta^{\\mathbf{k}} a^p \\right)_{\\mathbf{v}} = \\sum_{i \\in \\mathbb Z, \\, \\mathbf{w} \\in \\mathcal{V}} c^{p,i}_{\\mathbf{w}} \\left( \\Delta^{\\mathbf{k}} a^{p+i}\\right)_{\\mathbf{v}+\\mathbf{w}}. \\]\nTherefore, using the induction hypothesis and Item~\\ref{item_coeff_equal_one} above, we can write \n\\begin{align*}\n0 & \\leq \\left( \\Delta^{\\mathbf{k}} a^{p+2j-2} \\right)_{\\mathbf{v}+2\\cdot\\mathbf{1}_j}\\\\\n&= c^{p,2j-2}_{2\\cdot\\mathbf{1}_j} \\left( \\Delta^{\\mathbf{k}} a^{p+2j-2} \\right)_{\\mathbf{v}+2\\cdot\\mathbf{1}_j}\\\\\n&= \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}} - \\sum_{(i,\\mathbf{w}) \\ne (2j-2, 2\\cdot\\mathbf{1}_j)} c^{p,i}_{\\mathbf{w}} \\left( \\Delta^{\\mathbf{k}} a^{p+i} \\right)_{\\mathbf{v}+\\mathbf{w}}.\n\\end{align*}\nUsing the induction hypothesis again, we can remove all the terms in the last sum except the one where $(i,\\mathbf{w})=(0,\\mathbf{1}_j)$. Moreover, by Item~\\ref{item_coeff_positive} above, we can replace the coefficient $c^{p,0}_{\\mathbf{1}_j}$ by $1$. We obtain\n\\[0 \\leq \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}} - \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}+\\mathbf{1}_j}= \\left( \\Delta^{\\mathbf{k}+\\mathbf{1}_j} a^p \\right)_{\\mathbf{v}}, \\]\nwhich proves the induction step for $j \\geq 2$. If $j=1$, Item~\\ref{item_coeff_positive} is not true anymore (it is not possible to add only one face of degree $2$ in $2$ steps without changing the perimeter). Therefore, instead of \\eqref{eqn_double_peeling_equation}, we use the simple peeling equation \\eqref{eqn_peeling_equation_bipartite} like in \\cite{BL19}. More precisely, in the induction step, we fix $j' \\geq 2$ and write, using \\eqref{eqn_peeling_equation_bipartite}:\n\\begin{align*}\n0 &\\leq \\left( \\Delta^{\\mathbf{k}} a^{p+j'-1} \\right)_{\\mathbf{v}+\\mathbf{1}_{j'}}\\\\\n&= \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}} - \\sum_{j'' \\ne j'} \\left( \\Delta^{\\mathbf{k}} a^{p+j''-1} \\right)_{\\mathbf{v}+\\mathbf{1}_{j''}} -2 \\sum_{i=0}^{p-1} \\sum_{\\mathbf{w}} \\beta_0^{(i-1)}(\\mathbf{w}) \\left( \\Delta^{\\mathbf{k}} a^{p-i} \\right)_{\\mathbf{v}+\\mathbf{w}}.\n\\end{align*}\nEach term in the two sums is nonnegative by the induction hypothesis, so we can remove the second sum and keep only the term $j''=1$ in the first one to obtain\n\\[0 \\leq \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}} - \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}+\\mathbf{1}_1} = \\left( \\Delta^{\\mathbf{k}+\\mathbf{1}_1} a^{p} \\right)_{\\mathbf{v}}.\\]\nThis concludes the proof of the lemma.\n\\end{proof}\n\nBy Lemma~\\ref{lem_abs_monotone} and Proposition~\\ref{prop_moment_Hausdorff}, for all $p \\geq 1$, there is a $\\sigma$-finite measure $\\mu_p$ on $\\mathcal{Q}^*$ such that, for all $\\mathbf{v} \\in \\mathcal{V}^*_p$,\n\\begin{equation}\\label{eqn_apv_as_moment}\na^p_{\\mathbf{v}}=\\int_{\\mathcal{Q}^*} \\mathbf{q}^{\\mathbf{v}} \\mu_p(\\mathrm{d} \\q)\n\\end{equation}\nand furthermore $\\mu_1(\\mathcal{Q}^*) \\leq a^1_{\\mathbf{0}}=1$. We now replace $a_{\\mathbf{v}}^p$ by this expression in the peeling equation \\eqref{eqn_peeling_equation_bipartite}. We get\n\\begin{align*}\n\\int \\mathbf{q}^{\\mathbf{v}} \\, \\mu_p(\\mathrm{d} \\mathbf{q}) \\hspace{-0.1cm}&= \\hspace{-0.1cm}\\sum_{j \\geq 1} \\int \\mathbf{q}^{\\mathbf{v}+\\mathbf{1}_j} \\, \\mu_{p+j-1}(\\mathrm{d} \\q) + 2\\sum_{i=1}^{p-1} \\sum_{\\mathbf{w} \\in \\mathcal{V}} \\beta_0^{(i-1)}(\\mathbf{w}) \\int \\mathbf{q}^{\\mathbf{v}+\\mathbf{w}} \\, \\mu_{p-i}(\\mathrm{d} \\q)\\\\\n&=\\hspace{-0.1cm} \\int \\mathbf{q}^{\\mathbf{v}} \\left( \\sum_{j \\geq 1} q_j \\, \\mu_{p+j-1}(\\mathrm{d} \\q) + 2 \\sum_{i=1}^{p-1} W_{i-1}(\\mathbf{q}) \\, \\mu_{p-i}(\\mathrm{d} \\q) \\right),\n\\end{align*}\nwhere we recall that $W_{i-1}(\\mathbf{q})$ is the partition function of Boltzmann bipartite maps of the $2(i-1)$-gon with Boltzmann weights $\\mathbf{q}$.\nIn particular, the right-hand side for $i=2$ must be finite, which means that $\\mu_p$ is supported by the set $\\mathcal{Q}_a$ of admissible weight sequences. Moreover, the last display means that the two measures\n\\[ \\mu_p(\\mathrm{d} \\q) \\mbox{ and } \\nu_p(\\mathrm{d} \\q) = \\sum_{j \\geq 1} q_j \\, \\mu_{p+j-1}(\\mathrm{d} \\q) + 2 \\sum_{i=1}^{p-1} W_{i-1}(\\mathbf{q}) \\, \\mu_{p-i}(\\mathrm{d} \\q) \\]\nhave the same $\\mathbf{v}$-th moment for all $\\mathbf{v} \\in \\mathcal{V}^*_p$. In particular, if we fix $j \\geq 2$, this is true as soon as $v_j \\geq p$, so the measures $q_j^p \\mu_p(\\mathrm{d} \\q)$ and $q_j^p \\nu_p(\\mathrm{d} \\q)$ have the same moments so they are equal, so $\\mu_p$ and $\\nu_p$ coincide on $\\{q_j>0\\}$. Since this is true for all $j \\geq 2$ and $\\mu_p, \\nu_p$ are defined on $\\mathcal{Q}^*=\\bigcup_{j \\geq 2} \\{q_j>0\\}$, the measures $\\mu_p$ and $\\nu_p$ are the same, that is,\n\\begin{equation}\\label{peeling_equation_mu}\n\\mu_p(\\mathrm{d} \\q) = \\sum_{j \\geq 1} q_j \\, \\mu_{p+j-1}(\\mathrm{d} \\q) + 2 \\sum_{i=1}^{p-1} W_{i-1}(\\mathbf{q}) \\, \\mu_{p-i}(\\mathrm{d} \\q).\n\\end{equation}\nWe now note that this equation is very similar to the one satisfied by the constants $C_p(\\mathbf{q})$ used to define the $\\mathbf{q}$-IBPM. More precisely, we fix a finite measure $\\mu$ such that all the $\\mu_p$ are absolutely continuous with respect to $\\mu$ (take e.g. $\\mu (\\mathrm{d} \\q)=\\sum_{p \\geq 1} \\frac{g_p(\\mathbf{q}) \\mu_p(\\mathrm{d} \\q)}{2^p}$, where $g_p(\\mathbf{q})>0$ is such that the total mass of $g_p(\\mathbf{q}) \\mu_p(\\mathrm{d} \\q)$ is at most $1$). We denote by $f_p(\\mathbf{q})$ the density of $\\mu_p$ with respect to $\\mu$. Then \\eqref{peeling_equation_mu} becomes\n\\[ f_p(\\mathbf{q}) = \\sum_{j \\geq 1} q_j f_{p+j-1}(\\mathbf{q}) + 2 \\sum_{i=1}^{p-1} W_{i-1}(\\mathbf{q}) f_{p-i}(\\mathbf{q}) \\]\nfor $\\mu$-almost every $\\mathbf{q} \\in \\mathcal{Q}^*$. In other words, $\\left( f_p(\\mathbf{q}) \\right)_{p \\geq 1}$ satisfies the exact same equation as $\\left( C_p(\\mathbf{q}) \\right)_{p \\geq 1}$ in \\cite[Appendix C]{B18these}. These equations have a nonzero solution if and only if $\\mathbf{q} \\in \\mathcal{Q}_h$, so the measures $\\mu_p$ are actually supported by $\\mathcal{Q}_h$. Moreover, by uniqueness of the solution (up to a multiplicative constant), we have\n\\[ f_p(\\mathbf{q})=\\frac{C_p(\\mathbf{q})}{C_1(\\mathbf{q})} f_1(\\mathbf{q}) = C_p(\\mathbf{q}) f_1(\\mathbf{q})\\]\nfor $\\mu$-almost every $\\mathbf{q}$, so $\\mu_p(\\mathrm{d} \\q)=C_p(\\mathbf{q}) \\mu_1(\\mathrm{d} \\q)$. Now let $\\alpha \\leq 1$ be the total mass of the measure $\\mu_1$, and let $\\mathbf{Q}$ be a random variable with distribution $\\alpha^{-1} \\mu$. We then have, for all $p \\geq 1$ and $\\mathbf{v} \\in \\mathcal{V}^*_p$, if $m$ is a map with half-perimeter $p$ and face degrees $\\mathbf{v}$:\n\\begin{equation}\\label{eqn_apv_expectation}\n\\P \\left( m \\subset M \\right) = a^p_{\\mathbf{v}} = \\int \\mathbf{q}^{\\mathbf{v}} \\mu_p(\\mathrm{d} \\q) = \\alpha \\mathbb E \\left[ C_p(\\mathbf{Q}) \\mathbf{Q}^{\\mathbf{v}} \\right] = \\alpha \\P \\left( m \\subset \\mathbb M_{\\mathbf{Q}} \\right).\n\\end{equation}\nNote that $\\mathbf{Q}$ is not well-defined if $\\alpha=0$, but in this case $\\mu_p=0$ for all $p$ so \\eqref{eqn_apv_expectation} remains true for any choice of $\\mathbf{Q}$. To conclude that $M$ has the law of $\\mathbb M_{\\mathbf{Q}}$, all we have left to prove is that $\\alpha=1$ and that \\eqref{eqn_apv_expectation} can be extended to any $\\mathbf{v} \\in \\mathcal{V}_p$. For this, we will show that, when we explore $M$ via a peeling exploration, the perimeter and volumes of the explored region at time $t$ satisfy $\\mathbf{v} \\in \\mathcal{V}_p^*$ for $t$ large enough.\n\nMore precisely, if $\\mathcal{A}$ is a peeling algorithm, we recall that $\\mathcal{E}^{\\mathcal{A}}_t(M)$ is the explored part of $M$ after $t$ steps of a filled-in peeling exploration according to $\\mathcal{A}$. We denote by $P_t$ the half-perimeter of the hole of $\\mathcal{E}^{\\mathcal{A}}_t(M)$ and by $\\mathbf{V}_t$ the sequence of degrees of its internal faces (that is, $V_{t,j}$ is the number of internal faces of $\\mathcal{E}^{\\mathcal{A}}_t(M)$ with degree $2j$). Since $M$ is weakly Markovian, the process $\\left( P_t, \\mathbf{V}_t \\right)_{t \\geq 0}$ is a Markov chain whose law does not depend on the peeling algorithm $\\mathcal{A}$.\n\n\\begin{lem}\\label{lem_volumes_star}\nWe have\n\\[ \\P \\left( \\mathbf{V}_{t} \\in \\mathcal{V}^*_{P_t} \\right) \\xrightarrow[t \\to +\\infty]{} 1.\\]\n\\end{lem}\n\n\\begin{proof}\nSince the probability in the lemma does not depend on $\\mathcal{A}$, it is sufficient to prove the result for a particular peeling algorithm. Therefore, we can assume that $\\mathcal{A}$ has the following property: if the root face of $m$ and its hole have a common vertex $m$, then the peeled edge $\\mathcal{A}(m)$ is incident to such a vertex. We will prove that for this algorithm, we have a.s. $\\mathbf{V}_{t} \\in \\mathcal{V}^*_{P_t}$ for $t$ large enough.\n\nMore precisely, since the vertex degrees of $M$ are a.s. finite and by definition of $\\mathcal{A}$, all the vertices incident to the root face will eventually disappear from the boundary of the explored part. Therefore, for $t$ large enough, no vertex incident to the root face is on $\\partial \\mathcal{E}^{\\mathcal{A}}_t(M)$. We now fix $t$ with this property. If we denote by $\\mathrm{Inn}(m)$ the number of internal vertices of a map $m$ with a hole and by $2J$ the degree of the root face of $M$, this implies $\\mathrm{Inn} \\left( \\mathcal{E}^{\\mathcal{A}}_t(M) \\right) \\geq 2J$ for $t$ large enough.\n\nOn the other hand, the total number of edges of $\\mathcal{E}^{\\mathcal{A}}_t(M)$ is $p+\\sum_{j \\geq 1} j V_{t,j}$, so by the Euler formula\n\\begin{align*}\n\\mathrm{Inn} \\left( \\mathcal{E}^{\\mathcal{A}}_t(M) \\right) &= 2 + \\left( P_t+ \\sum_{j \\geq 1} j V_{t,j} \\right) - \\left( 1+\\sum_{j \\geq 1} V_{t,j} \\right) -2P_t \\\\&= 1-P_t+\\sum_{j \\geq 1} (j-1) V_{t,j}. \n\\end{align*}\nTaking $t$ large enough to have $\\mathrm{Inn} \\left( \\mathcal{E}^{\\mathcal{A}}_t(M) \\right) \\geq 2J$, we obtain\n\\[ \\left( \\sum_{j \\geq 1} (j-1) V_{t,j} \\right) -(J-1) \\geq \\left( 2J +P_t -1 \\right) -(J-1) = P_t+J > P_t-1,\\]\nso $V_{t,J}>0$ and $\\mathbf{V}-\\mathbf{1}_{J} \\in \\mathcal{V}_{P_t}$. This proves $\\mathbf{V}_{t} \\in \\mathcal{V}^*_{P_t}$ for $t$ large enough.\n\\end{proof}\n\nWe now conclude the proof of Theorem~\\ref{thm_weak_Markov_general} from \\eqref{eqn_apv_expectation}. We consider a finite map $m_0$ with a hole and a peeling algorithm $\\mathcal{A}$ that is consistent with $m_0$ in the sense that $m_0$ is a possible value of $\\mathcal{E}_{t_0}^{\\mathcal{A}}$ for some $t_0 \\geq 0$. We note that $\\mathcal{E}^{\\mathcal{A}}_{t_0}(M)=m_0$ if and only if $m_0 \\subset M$. Indeed, the direct implication is immediate. The indirect one comes from the fact that, if $m_0 \\subset M$, then all the peeling steps until time $t_0$ must be consistent with $m_0$, so $m_0 \\subset M$ determines the first $t_0$ peeling steps. We now take $t \\geq t_0$. We sum \\eqref{eqn_apv_expectation} over all possible values $m$ of $\\mathcal{E}^{\\mathcal{A}}_t(M)$ such that $m_0 \\subset m$ and the half-perimeter $p$ and internal face degrees $\\mathbf{v}$ of $m$ satisfy $\\mathbf{v} \\in \\mathcal{V}_p^*$. We get\n\\[ \\P \\left( m_0 \\subset M \\mbox{ and } \\mathbf{V}_t \\in \\mathcal{V}^*_{P_t} \\right) = \\alpha \\P \\left( m_0 \\subset \\mathbb M_{\\mathbf{Q}} \\mbox{ and } \\mathbf{V}^{\\mathbf{Q}}_t \\in \\mathcal{V}^*_{P_t^{\\mathbf{Q}}} \\right), \\]\nwhere $P_t^{\\mathbf{Q}}$ and $\\mathbf{V}_t^{\\mathbf{Q}}$ are the analogues of $P_t$ and $\\mathbf{V}_t$ for $\\mathbb M_{\\mathbf{Q}}$ instead of $M$. Since $\\mathbb M_{\\mathbf{Q}}$ is weakly Markovian, we can apply Lemma~\\ref{lem_volumes_star} to both $M$ and $\\mathbb M_{\\mathbf{Q}}$. Therefore, letting $t \\to +\\infty$ in the last display, we get \\[\\P \\left( m_0 \\subset M \\right) = \\alpha \\P \\left( m_0 \\subset \\mathbb M_{\\mathbf{Q}} \\right)\\] for all $m_0$. In particular, if $m_0$ is the trivial map consisting only of the root edge, we get $\\alpha=1$, so $M$ and $\\mathbb M_{\\mathbf{Q}}$ have the same law. This proves Theorem~\\ref{thm_weak_Markov_general}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm_main_more_general}]\nBy Proposition~\\ref{prop_tightness_dloc_univ}, any subsequential limit $M$ of $\\left( M_{\\mathbf{f}^n, g_n} \\right)$ is planar and one-ended. Moreover, let $m$ be a map with one hole of half-perimeter $p$ and $v_{j}$ faces of degree $2j$ for all $j \\geq 1$. Then\n\\[\\P \\left( m \\subset M \\right) = \\lim_{n \\to +\\infty} \\P \\left( m \\subset M_{\\mathbf{f}^n, g_n} \\right) = \\lim_{n \\to +\\infty} \\frac{\\beta_{g_n}^{(p)}(\\mathbf{f}^n-\\mathbf{v})}{\\beta_{g_n}(\\mathbf{f}^n)},\\]\nwhere the limits are along some subsequence. In particular, the dependence in $m$ is only in $p$ and $\\mathbf{v}$, so $M$ is weakly Markovian and the result follows by Theorem~\\ref{thm_weak_Markov_general}.\n\\end{proof}\n\n\\section{The parameters are deterministic}\\label{sec_univ_end}\n\n\\subsection{Outline}\\label{sec_arg_deux_trous}\n\nOur goal is now to prove Theorem~\\ref{univ_main_thm}. We fix face degree sequences $\\mathbf{f}^n$ and genuses $g_n$ for $n \\geq 0$ satisfying the assumptions of Theorem~\\ref{univ_main_thm} (in particular, we now assume $\\sum_{j} j^2 \\alpha_j <+\\infty$ until the end of the paper). By Theorem~\\ref{thm_main_more_general}, up to extracting a subsequence, we can assume $M_{\\mathbf{f}^n, g_n}$ converges to $\\mathbb M_{\\mathbf{Q}}$, where $\\mathbf{Q}$ is a random variable with values in $\\mathcal{Q}_h$. Moreover, the law of the degree of the root face in $M_{\\mathbf{f}^n, g_n}$ converges in distribution to $\\left( j \\alpha_j \\right)_{j \\geq 1}$, which has finite expectation. By the last point of Theorem~\\ref{thm_weak_Markov_general}, we have $\\mathbf{Q} \\in \\mathcal{Q}_f$ a.s.. To prove Theorem~\\ref{univ_main_thm}, it is enough to prove that $\\mathbf{Q}$ is deterministic, and only depends on $\\left( \\alpha_j \\right)_{j \\geq 1}$ and $\\theta$.\n\n\\paragraph{Sketch of the end of the proof.}\nSince we will follow similar ideas, let us first recall the strategy of \\cite{BL19}. If $e_n$ is the root edge of $M_{\\mathbf{f}^n, g_n}$, the parameters $\\mathbf{Q}$ can be observed on a large neighbourhood of $e_n$ in $M_{\\mathbf{f}^n, g_n}$ for $n$ very large.\nThe first step of the proof (Proposition~\\ref{prop_two_holes_argument}) roughly consists of showing that once $M_{\\mathbf{f}^n, g_n}$ is picked, the weights $\\mathbf{Q}$ do not depend on the choice of $e_n$. This is proved by the \\emph{two holes argument}: if $e^1_n$ and $e_n^2$ are two roots chosen uniformly on $M_{\\mathbf{f}^n, g_n}$, we swap two large neighbourhoods of $e^1_n$ and $e^2_n$ in $M_{\\mathbf{f}^n, g_n}$. We then remark that if the weights observed around the two roots are too different, then the map obtained after swapping does not look like a map of the form $\\mathbb M_{\\mathbf{Q}}$.\nThe second step consists of noticing that the average value over all choices of the root of some functions of $\\mathbf{Q}$ is fixed by $\\left( \\alpha_j \\right)_{j \\geq 1}$ and $\\theta$ (Corollary~\\ref{corr_main_minus_monotonicity}).\nFinally, in the third step we prove that these functions are sufficient to characterize $\\mathbf{Q}$ completely (Proposition~\\ref{prop_monotonicity_deg}).\n\nHowever, two important difficulties appear here compared to~\\cite{BL19}:\n\\begin{itemize}\n\\item\nin the first step, we need to find two large pieces around $e_n^1$ and $e_2^n$ with the exact same perimeter, in order to be able to swap them. This was easy for triangulations since the perimeter process associated to a peeling exploration takes all values. This is not true anymore in our general setting. This part will make crucial use of the assumption $\\sum_j j^2 \\alpha_j < +\\infty$ (this is actually the only place in the paper where we will use it). A consequence of this difficulty is that instead of performing the swapping operation \\emph{with high probability}, we will perform it \\emph{with positive probability}.\n\\item\nIn the third step, one of the parameters that we control is the average vertex degree. For triangulations, it followed from an explicit formula that the average degree characterizes $\\mathbf{Q}$. We do not have such a formula here, so our argument will be more involved, and rely on the partial results obtained so far in the present paper.\n\\end{itemize}\n\n\\paragraph{Intermediate results.}\nLet $\\left( M_n, e_n^1, e_n^2 \\right)$ be a uniform, bi-rooted map with face degrees $\\mathbf{f}^n$ and genus $g_n$ (i.e. $e_n^1$ and $e_n^2$ are picked uniformly and independently among the edges of $M_n$). We highlight that we will write $M_n$ instead of $M_{\\mathbf{f}^n, g_n}$ in this section to make notations lighter.\nUp to extracting a subsequence, we can assume the joint convergence\n\\[ \\left( (M_n, e_n^1), (M_n, e_n^2) \\right) \\xrightarrow[n \\to +\\infty]{(d)} \\left( \\mathbb M^1_{\\mathbf{Q}^1}, \\mathbb M^2_{\\mathbf{Q}^2} \\right) \\]\nfor the local topology, where $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$ have the same distribution as $\\mathbf{Q}$. Moreover, by the Skorokhod representation theorem, we can assume this joint convergence is almost sure. \\emph{We will stay in this setting in Section~\\ref{subsec_same_perimeter} and~\\ref{subsec_two_holes}.} The first step of the proof will consist of proving the following.\n\n\\begin{prop}\\label{prop_two_holes_argument}\nWe have $\\mathbf{Q}^1=\\mathbf{Q}^2$ almost surely.\n\\end{prop}\n\nFrom here, the second step will be to deduce the next result. For $\\mathbf{q} \\in \\mathcal{Q}_h$, we recall that $j a_j(\\mathbf{q})$ is the probability that the root face of $\\mathbb M_{\\mathbf{q}}$ has degree $2j$, and that $d(\\mathbf{q})=\\mathbb E \\left[ \\frac{1}{\\mathrm{deg}_{\\mathbb M_{\\mathbf{q}}} (\\rho)} \\right]$. \n\n\\begin{corr}\\label{corr_main_minus_monotonicity}\nUnder the assumptions of Theorem~\\ref{univ_main_thm}, let $\\mathbb M_{\\mathbf{Q}}$ be a subsequential limit. Then almost surely, we have\n\\[ d(\\mathbf{Q})=\\frac{1}{2} \\left( 1-2\\theta-\\sum_i \\alpha_i \\right) \\mbox{ and, for all $j \\geq 1$, } a_j(\\mathbf{Q})=\\alpha_j.\\] \n\\end{corr}\n\n\\paragraph{Structure of the section.}\nIn Section~\\ref{subsec_same_perimeter}, we will address the issue of finding two large neighbourhoods of the two roots with exactly the same perimeters. In Section~\\ref{subsec_two_holes}, we use this to prove Proposition~\\ref{prop_two_holes_argument} and Corollary~\\ref{corr_main_minus_monotonicity}. Finally, Section~\\ref{subsec_last_step} is devoted to the end of the proof of Theorem~\\ref{univ_main_thm}, and consists mostly of showing that $d(\\mathbf{q})$ and $\\left( a_j(\\mathbf{q}) \\right)_{j \\geq 1}$ are sufficient to characterize $\\mathbf{q}$.\n\n\\subsection{Finding two pieces with the same perimeter}\n\\label{subsec_same_perimeter}\n\nAs explained above, given the uniform bi-rooted map $\\left( M_n, e^1_n, e^2_n \\right)$, we want to find two neighbourhoods of $e^1_n$ and $e^2_n$ with the same large perimeter $2p$. For this, we will perform a peeling exploration around the two roots and stop it when the perimeter of the explored region is exactly $2p$. However, since the perimeter process has a positive drift, it can make large positive jumps, we cannot guarantee that both perimeters will hit the value $p$ with high probability. We will therefore show a weaker result: roughly speaking, the probability that the perimeter processes around $e_n^1$ and $e_n^2$ both hit $p$ is bounded from below, even if we condition on $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$. \n\nMore precisely, we fix a deterministic peeling algorithm $\\mathcal{A}$, and let $p,v_0 \\geq 1$. We recall from the end of Section~\\ref{subsec_lazy_peeling} that we can make sense of a filled-in peeling exploration on the finite map $M_n$ around $e^1_n$ or $e_n^2$. We perform the following exploration:\n\\begin{itemize}\n\\item\nwe explore the map $M_n$ around $e^1_n$ according to the algorithm $\\mathcal{A}$ until the number of edges in the explored region is larger than $v_0$, or the perimeter of the explored region is exactly $2p$, and denote by $\\tau^1_n$ the time at which we stop;\n\\item\nwe do the same thing around $e^2_n$ and denote by $\\tau^2_n$ the stopping time.\n\\end{itemize}\nWe write $\\mathcal{S}_{n,p,v_0}$ for the event where both $\\tau^1_n$ and $\\tau^2_n$ occur because the perimeter hits $2p$, and where the two regions explored around $e_n^1$ and $e_n^2$ are face-disjoint (the dependence of $\\mathcal{S}$ in $\\mathcal{A}$ will stay implicit). We note right now that $(M_n, e_n^1)$ has a planar, one-ended local limit. Hence, with probability $1-o(1)$ as $n \\to +\\infty$, the exploration is not stopped before $\\tau_n^1$ or $\\tau_n^2$ for the reason stated in the end of Section~\\ref{subsec_lazy_peeling}.\n\nThe goal of this subsection is to prove the next result. We recall that the functions $r_j(\\mathbf{q})$ for $j \\in \\mathbb N^* \\cup \\{ \\infty \\}$ and $\\mathbf{q} \\in \\mathcal{Q}_h$ are defined in Proposition~\\ref{prop_q_as_limit}.\n\n\\begin{prop}\\label{prop_finding_regions_to_cut}\nLet $(M_n, e^1_n, e_2^n)$ and $\\mathbf{Q}^1, \\mathbf{Q}^2$ be as in Section~\\ref{sec_arg_deux_trous}.\nWe fix $j \\in \\mathbb N^* \\cup \\{ \\infty \\}$, and $\\varepsilon>0$. Then there is $\\delta>0$ with the following property. For every $p \\geq 1$ large enough, there is $v_0$ such that, for $n$ large enough:\n\\[ \\mbox{if } \\P \\left( | r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2) |>\\varepsilon \\right) \\geq \\varepsilon,\\]\n\\[ \\mbox{then }\\P \\left( | r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2)|>\\frac{\\varepsilon}{2} \\mbox{ and } (M_n, e_n^1, e_n^2) \\in \\mathcal{S}_{n,p,v_0} \\right) \\geq \\delta. \\]\n\\end{prop}\n\nWe recall that we have used the Skorokhod theorem to couple the finite and infinite maps together, so the last event makes sense.\n\nHere is why Proposition~\\ref{prop_finding_regions_to_cut} seems reasonable: we know that conditionally on $(\\mathbf{Q}^1, \\mathbf{Q}^2)$, the perimeters of the explored region along a peeling exploration of $\\mathbb M^1_{\\mathbf{Q}^1}$ and $\\mathbb M^2_{\\mathbf{Q}^2}$ are random walks conditioned to stay positive. Moreover, since $\\mathbf{Q}^1, \\mathbf{Q}^2 \\in \\mathcal{Q}_f$, these random walks do not have a too heavy tail, so each of them have a reasonable chance of hitting exactly $p$. However, there is no reason a priori why $\\mathbb M^1_{\\mathbf{Q}^1}$ and $\\mathbb M^2_{\\mathbf{Q}^2}$ should be independent conditionally on $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$, so it might be unlikely that \\emph{both} processes hit $p$. Therefore, the sketch of the proof will be the following:\n\\begin{itemize}\n\\item\nwe fix a large constant $C>0$ ($C$ will be much smaller than $p$),\n\\item\nwe prove that both walks have a large probability to hit the interval $[p,p+C]$ before the explored volume exceeds $v_0(p)$ (Lemma~\\ref{lem_RW_hits_interval}),\n\\item\nonce both perimeter processes around $e_n^1$ and $e_n^2$ in $M_n$ have hit $[p,p+C]$, we use the Bounded ratio Lemma (Lemma~\\ref{lem_BRL}, item 2) to show that, with probability bounded from below by roughly $e^{-C}$, both perimeters fall to exactly $p$ in at most $C$ steps. This will prove the proposition with $\\delta \\approx \\varepsilon e^{-C}$ and $v_0=v_0(p)$.\n\\end{itemize}\nThe point of replacing $p$ by $[p,p+C]$ is to deal with events of large probability, so that we don't need any independence to make sure that two events simultaneously happen.\n\nFor this, consider the peeling exploration of $\\mathbb M^1_{\\mathbf{Q}^1}$ according to $\\mathcal{A}$. We denote by $\\sigma^{1,\\infty}_{[p,p+C]}$ the first time at which the half-perimeter is in $[p,p+C]$ (this stopping time might be infinite). We define $\\sigma^{1,n}_{[p,p+C]}$ (resp. $\\sigma^{2, \\infty}_{[p,p+C]}$, $\\sigma^{2,n}_{[p,p+C]}$) as the analogue quantity for the exploration in $M_n$ around $e_n^1$ (resp. in $\\mathbb M^2_{\\mathbf{Q}^2}$, in $\\left( M_n, e_n^2 \\right)$).\n\n\\begin{lem}\\label{lem_RW_hits_interval}\nWe have\n\\[ \\lim_{C \\to +\\infty} \\liminf_{p \\to +\\infty} \\P \\left( \\sigma_{[p,p+C]}^{1,\\infty} <+\\infty \\right) =1.\\]\n\\end{lem}\n\n\\begin{proof}\nWe know that $\\mathbf{Q}^1 \\in \\mathcal{Q}_f$ a.s.. Hence, it is enough to prove that, for any $\\mathbf{q} \\in \\mathcal{Q}_f$, we have\n\\begin{equation}\\label{eqn_RW_hits_interval_infinite}\n\\lim_{C \\to +\\infty} \\liminf_{p \\to +\\infty} \\P \\left( \\sigma^{1,\\infty}_{[p,p+C]}<+\\infty \\big| \\mathbf{\\mathbf{Q}}=\\mathbf{q} \\right) =1.\n\\end{equation}\nThe lemma then follows by taking the expectation and using Fatou's lemma. Conditionally on $\\mathbf{Q}^1=\\mathbf{q}$, the law of $\\mathbb M^1_{\\mathbf{Q}^1}$ is the law of $\\mathbb M_\\mathbf{q}$. In particular, the process $P$ describing the half-perimeter of the explored region has the law of $X$ conditioned to stay positive, where $X$ is a random walk with step distribution $\\widetilde{\\nu}_{\\mathbf{q}}$.\n\nTo prove \\eqref{eqn_RW_hits_interval_infinite}, we distinguish two cases: the case where $\\mathbf{q}$ is critical, and the case where it is not. We start with the second one. Then by the results of Section~\\ref{subsec_IBPM}, the walk $X$ satisfies $\\mathbb E \\left[ |X_1| \\right]<+\\infty$ and $\\mathbb E \\left[ X_1 \\right]>0$, so the conditioning to stay positive is non degenerate. Therefore, it is enough to prove\n\\begin{equation}\\label{eqn_RW_hits_interval_conditioned}\n\\lim_{C \\to +\\infty} \\liminf_{p \\to +\\infty} \\P \\left( \\mbox{$X$ hits $[p,p+C]$} \\right) =1.\n\\end{equation}\nThis follows from standard renewal arguments: if we denote by $(H_i)_{i \\geq 0}$ the ascending ladder heights of $P$, then $(H_i)$ is a renewal set with density $\\mathbb E[H_1]= \\frac{1}{\\mathbb E[X_1]}>0$. Let $I_p$ be such that $H_{I_p} < p \\leq H_{I_{p+1}}$. Then the law of $H_{I_{p+1}}-H_{I_p}$ converges as $p \\to +\\infty$ to the law of $H_1$ biased by its size, so\n\\begin{align*}\n\\P \\left( \\mbox{$P$ does not hit $[p,p+C]$} \\right) &\\leq \\P \\left( H_{I_{p+1}} \\notin [p,p+C] \\right)\\\\& \\leq \\P \\left( H_{I_{p+1}}-H_{I_p} > C \\right) \\xrightarrow[p \\to +\\infty]{} \\frac{\\mathbb E \\left[ H_1 \\mathbbm{1}_{H_1>C} \\right]}{\\mathbb E[H_1]},\n\\end{align*} \nand this last quantity goes to $0$ as $C \\to +\\infty$.\n\nWe now tackle the case where $\\mathbf{q}$ is critical, which by the results in the end of~\\ref{subsec_IBPM} implies $\\sum_{i \\geq 1} i^{3\/2} \\widetilde{\\nu}_{\\mathbf{q}}(i)<+\\infty$. This case is more complicated since renewal arguments are not available anymore, and the conditioning is now degenerate, so absolute continuity arguments between $P$ and $X$ become more elaborate. On the other hand, the growth is now slower and the nonconditionned walk $X$ with step distribution $\\widetilde{\\nu}_{\\mathbf{q}}$ is now recurrent, so it seems more difficult to jump over a large interval. And indeed, we will prove\n\\[ \\lim_{p \\to +\\infty} \\P \\left( \\mbox{$P$ hits $p$} \\right) =1,\\]\nwhich is a much stronger version of \\eqref{eqn_RW_hits_interval_conditioned}.\n\nFor this, our strategy will be the following: let $\\chi_p$ be the first time at which $P$ is at least $p$.\n\\begin{itemize}\n\\item\nThe scaling limit of $P$ is a process with no positive jump, so ${P_{\\chi_p}=p+o(p)}$ in probability as $p \\to +\\infty$.\n\\item\nBetween time $\\chi_p$ and $\\tau_p+o(p^{2\/3})$, the process $P$ looks a lot like a nonconditioned random walk $X$ started from $P_{\\tau_p}$.\n\\item\nIf $X$ is started from $p+o(p)$, the time it takes to first hit $P$ is $o(p^{2\/3})$. This is a stronger version of the recurrence of $X$, and will follow from a local limit theorem for random walks.\n\\end{itemize}\nLet us now be more precise. By Theorem 3 of \\cite{Bud15} (see also \\cite[Chapter 10]{C-StFlour}), we have the convergence\n\\[ \\left( \\frac{P_{nt}}{n^{2\/3}} \\right)_{t \\geq 0} \\xrightarrow[n \\to +\\infty]{(d)} \\left( b_{\\mathbf{q}} S^+_t \\right)_{t \\geq 0} \\]\nfor the Skorokhod topology, where $S^+$ is a $3\/2$-stable L\u00e9vy process with no positive jump conditioned to stay positive, and $b_{\\mathbf{q}}>0$ (the precise value will not matter here). Since this limiting process has no positive jump, we have $P_{\\tau_p}-p=o(p)$ in probability. Hence, there is a deterministic function $f(p)$ with $\\frac{f(p)}{p} \\to 0$ when $p \\to +\\infty$ such that, for any $\\varepsilon>0$,\n\\[ \\P \\left( P_{\\tau_p}-p \\geq \\varepsilon f(p) \\right) \\xrightarrow[p \\to +\\infty]{} 0. \\]\nWe now fix $\\varepsilon>0$, and condition on $P_{\\tau_p}=p'$ for some $p \\leq p' \\leq p+\\varepsilon f(p)$. We claim that then $\\left( P_{\\tau_p+i}-p' \\right)_{0 \\leq i \\leq f(p)^{3\/2}}$ can be coupled with $(X_i)_{0 \\leq i \\leq f(p)^{3\/2}}$ in such a way that both processes are the same with probability $1-o(1)$. For this, recall from~\\eqref{peeling_transitions} that $P$ can be described as a Doob $h$-transform of $X$, where $h$ is given by \\eqref{eqn_defn_homega}. Hence, the Radon--Nikodym derivative of the first process with respect to the second is\n\\begin{equation}\\label{critical_Radon_Nikodym}\n\\frac{h_{p'+X_{f(p)^{3\/2}}}(1)}{h_{p'}(1)}.\n\\end{equation}\nSince $\\frac{X_{f(p)^{3\/2}}}{f(p)}$ converges in distribution, we have $\\frac{X_{f(p)^{3\/2}}}{p} \\to 0$ in probability. By using the fact that $\\frac{p'}{p} \\to 0$ uniformly in $p'$ and that $h_1(x)\\sim c\\sqrt{x}$ for some $c>0$ (see Section~\\ref{subsec_IBPM}), we conclude that \\eqref{critical_Radon_Nikodym} goes to $1$ as $p \\to +\\infty$, uniformly in $p' \\in \\left[ p,p+\\varepsilon f(p) \\right]$. This proves our coupling claim. Note that under this coupling, the time where $P$ hits exactly $p$ is $\\tau_p$ plus the time where $X$ hits $p-p'$.\n\nWe will now show that, if $p$ is large enough, for any $k \\in [-\\varepsilon f(p),0]$, we have\n\\begin{equation}\\label{eqn_critical_quantitative_recurrence}\n\\P \\left( \\mbox{$X$ hits $k$ before time $f(p)^{3\/2}$} \\right) \\geq 1-\\delta(\\varepsilon),\n\\end{equation}\nwhere $\\delta(\\varepsilon) \\to 0$ as $\\varepsilon \\to 0$. Together with our coupling result, this will imply that the probability for $P$ to hit $p$ before time $\\tau_p+f(p)^{3\/2}$ is at least $1-\\delta (\\varepsilon)-o(1)$ as $p \\to +\\infty$. Since this is true for any $\\varepsilon>0$, this will conclude the proof of Lemma~\\ref{lem_RW_hits_interval} in the critical case.\n\nThe proof of \\eqref{eqn_critical_quantitative_recurrence} relies on the Local Limit Theorem (this is e.g. Theorem 4.2.1 of \\cite{IL71}). This theorem (in the case $\\alpha=3\/2$) states that\n\\[ \\sup_{k \\in \\mathbb Z} \\left| n^{2\/3} \\P (X_n=k)-g \\left( \\frac{k}{n^{2\/3}} \\right) \\right| \\xrightarrow[n \\to +\\infty]{} 0, \\]\nwhere $g$ is a continuous function (the density of a $3\/2$-stable variable).\nOn the other hand, let us denote $t=f(p)^{3\/2}$. By the strong Markov property, for all $k \\in \\mathbb Z$, we have\n\\begin{align*}\n\\mathbb E_0 \\left[ \\sum_{i=0}^{t} \\mathbbm{1}_{X_i=k} \\right] &\\leq \\P_0 \\left( \\mbox{$X$ hits $k$ before time $t$} \\right) \\mathbb E_k \\left[ \\sum_{i=0}^{t} \\mathbbm{1}_{X_i=k} \\right]\\\\\n&= \\P_0 \\left( \\mbox{$X$ hits $k$ before time $t$} \\right) \\mathbb E_0 \\left[ \\sum_{i=0}^{t} \\mathbbm{1}_{X_i=0} \\right].\n\\end{align*}\nTherefore, using the local limit theorem, we can write, for $-\\varepsilon f(p) \\leq k \\leq 0$ and $p$ large (the $o$ terms are all uniform in $k$):\n\\begin{align*}\n\\P_0 \\left( \\mbox{$X$ hits $k$ before $t$} \\right) &\\geq \\frac{\\sum_{i=0}^{t} \\P_0 \\left( X_i=k \\right)}{\\sum_{i=0}^{t} \\P_0 \\left( X_i=0 \\right)}\\\\\n&= \\frac{\\sum_{i=1}^{t} \\left( \\frac{1}{i^{2\/3}} g \\left( \\frac{k}{i^{2\/3}} \\right) + o \\left( \\frac{1}{i^{2\/3}} \\right) \\right)}{1+\\sum_{i=1}^{t} \\left( \\frac{1}{i^{2\/3}} g (0) + o \\left( \\frac{1}{i^{2\/3}} \\right) \\right)}\\\\\n&\\geq \\frac{-\\varepsilon t^{1\/3} + \\sum_{i=\\varepsilon t}^{t} \\left( \\frac{1}{i^{2\/3}} \\min_{[-\\varepsilon^{1\/3},0]} g +o \\left( \\frac{1}{i^{2\/3}} \\right)\\right)}{3t^{1\/3} g(0) + \\varepsilon t^{1\/3}}\\\\\n&\\geq \\frac{-2\\varepsilon t^{1\/3}+ \\left( 3t^{1\/3}-3\\varepsilon^{1\/3} t^{1\/3}\\right) \\min_{[-\\varepsilon^{1\/3},0]} g}{\\left( 3g(0)+\\varepsilon \\right)t^{1\/3}}\\\\\n&= \\frac{-2\\varepsilon +3(1-\\varepsilon^{1\/3}) \\min_{[-\\varepsilon^{1\/3},0]} g}{3g(0)+\\varepsilon},\n\\end{align*}\nwhere the third line uses that, for any index $i \\geq \\varepsilon t$, we have\n\\[0 \\geq \\frac{k}{i^{2\/3}} \\geq -\\frac{\\varepsilon f(p)}{(\\varepsilon t)^{2\/3}}=-\\varepsilon^{1\/3}. \\]\nWe obtain a lower bound that goes to $1$ as $\\varepsilon \\to 0$, so this proves \\eqref{eqn_critical_quantitative_recurrence}, and Lemma~\\ref{lem_RW_hits_interval}.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_finding_regions_to_cut}]\nThe subtlety in the proof is that we would like to say something about the finite maps $M_n$ conditionally on the values of $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$, but $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$ are defined in terms of the infinite limits. However, we can condition on the maps explored at the time when the perimeters of the explored parts hit $[p,p+C]$ for the first time. Then Proposition~\\ref{prop_q_as_limit} guarantees that from these explored parts, we can get good approximations of $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$ if $p$ is large enough.\n\nIn this proof, we will use a shortened notation for our peeling explorations. For $i \\in \\{1,2\\}$ and $t \\geq 0$, we will write $\\mathcal{E}^{n,i}_t=\\mathcal{E}_t^{\\mathcal{A}} \\left( M_n, e_n^i \\right)$ and $\\mathcal{E}^{\\infty,i}_t=\\mathcal{E}_t^{\\mathcal{A}} \\left( \\mathbb M^i_{\\mathbf{Q}^i} \\right)$.\n\nWe fix $\\varepsilon>0$. By Lemma~\\ref{lem_RW_hits_interval}, let $C$ be a constant (depending only on $\\varepsilon$) such that\n\\[ \\liminf_{p \\to +\\infty} \\P \\left( \\sigma_{[p,p+C]}^{1,\\infty} <+\\infty \\right) > 1-\\frac{\\varepsilon}{20}.\\]\nFor $p$ large enough (where \"large enough\" may depend on $\\varepsilon$), there is $v_0=v_0(p)$ such that\n\\begin{equation}\\label{eqn_sigma_reasonable}\n\\P \\left( \\sigma^{\\infty,1}_{[p,p+C]} \\leq v_0 \\mbox{ and } \\left| \\mathcal{E}^{\\infty,1}_{\\sigma^{\\infty,1}_{[p,p+C]}} \\right| \\leq v_0 \\right)>1-\\frac{\\varepsilon}{20},\n\\end{equation}\nwhere $|m|$ is the number of edges of a map $m$.\nOn the other hand, let us fix $j \\in \\mathbb N^* \\cup \\{\\infty\\}$. Proposition~\\ref{prop_q_as_limit} provides a function $\\widetilde{r}_j$ on the set of finite maps with a hole such that $\\widetilde{r}_j(\\mathcal{E}^{\\infty, 1}_t) \\to r_j(\\mathbf{Q}^1)$ almost surely as $t \\to +\\infty$. Let $\\eta<1$ be a small constant, which will be fixed later and will only depend on $\\varepsilon$. For $p$ large enough, we have\n\\begin{equation}\\label{eqn_fq_approximation}\n\\P \\left( \\sigma^{\\infty,1}_{[p,p+C]} \\leq v_0(p) \\mbox{ but } \\left| \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 1}_{\\sigma^{\\infty,1}_{[p,p+C]}} \\right) - r_j(\\mathbf{Q}^1) \\right| \\geq \\frac{\\varepsilon}{8} \\right) < \\eta \\frac{\\varepsilon}{20}.\n\\end{equation}\nFrom now on, we take $p$ large enough so that both \\eqref{eqn_sigma_reasonable} and \\eqref{eqn_fq_approximation} hold. By almost sure local convergence and \\eqref{eqn_sigma_reasonable}, for $n$ large enough (where \"large enough\" may depend on $\\varepsilon$ and $p$), we have\n\\[ \\P \\left( \\sigma^{n,1}_{[p,p+C]}, \\sigma^{n,2}_{[p,p+C]} \\leq v_0 \\mbox{ and } \\left| \\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}} \\right|, \\left| \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}} \\right| \\leq v_0 \\right) > 1-\\frac{\\varepsilon}{10}.\\]\nBy the assumption that $|r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2)|>\\varepsilon$ with probability at least $\\varepsilon$ and by \\eqref{eqn_fq_approximation}, we deduce that\n\\[ \\P \\left(\\begin{array}{c} \\sigma^{n,1}_{[p,p+C]}, \\sigma^{n,2}_{[p,p+C]} \\leq v_0 \\mbox{ and } \\left| \\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}} \\right|, \\left| \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}} \\right| \\leq v_0\\\\ \\mbox{and } \\left| \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 1}_{\\sigma^{\\infty,1}_{[p,p+C]}} \\right) - \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 2}_{\\sigma^{\\infty,2}_{[p,p+C]}} \\right) \\right| \\geq \\frac{3}{4}\\varepsilon\\end{array} \\right) > \\frac{4}{5} \\varepsilon.\\]\n\n\n\nNote that if this last event occurs but the two regions $\\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}}$ and $\\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}}$ have a common face, then the dual graph distance between the two roots is bounded by $2v_0$. However, by Proposition~\\ref{prop_tightness_dloc_univ}, the volume of the ball of radius $2v_0$ around $e^1_n$ is tight as $n \\to +\\infty$, so the probability that this happens goes to $0$ as $n \\to +\\infty$. Hence, for $n$ large enough:\n\\begin{equation}\\label{eqn_two_distinct_regions}\n\\P \\left( \\begin{array}{c}\\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}}, \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}} \\mbox{ are well-defined, face-disjoint, have at}\\\\\\mbox{most $v_0$ edges and } \\left| \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 1}_{\\sigma^{\\infty,1}_{[p,p+C]}} \\right) - \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 2}_{\\sigma^{\\infty,2}_{[p,p+C]}} \\right) \\right| \\geq \\frac{3}{4}\\varepsilon \\end{array} \\right) > \\frac{4}{5} \\varepsilon.\n\\end{equation}\n\nNow assume that this last event occurs and condition on the $\\sigma$-algebra $\\mathcal{F}_{\\sigma}$ generated by the pair $\\left( \\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}}, \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}} \\right)$ of explored regions. Then, let $I_1, I_2 \\in [0,C]$ be such that the perimeters of the two explored regions are $2p+2I_1$ and $2p+2I_2$. Then the complementary map is a uniform map of the $\\left( 2p+2I_1, 2p+2I_2 \\right)$-gon with genus $g_n$ and face degrees given by $\\mathbf{\\widetilde{F}}^n$ as follows. If $F_j^n$ is the number of internal faces of degree $2j$ in $\\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}} \\cup \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}}$, then $\\widetilde{F}_j^n=f_j^n-F_j^n$. We now perform $I_1$ peeling steps according to $\\mathcal{A}$ around $\\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}}$, followed by $I_2$ peeling steps according to $\\mathcal{A}$ around $\\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}}$. We call a peeling step \\emph{nice} if it consists of gluing together two boundary edges, which decreases the perimeter by $2$. The number of possible values of the map $M_n \\backslash \\left( \\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}} \\cup \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}} \\right)$ is\n\\[\\beta_{g}^{(p+I_1,p+I_2)}(\\mathbf{\\widetilde{F}}^n).\\]\nOn the other hand, if the $I_1+I_2$ additional peeling steps are all good and the regions around $e_n^1$ and $e_n^2$ are still disjoint after these steps, the number of possible complementary maps is\n\\[\\beta_{g}^{(p,p)} (\\mathbf{\\widetilde{F}}^n). \\]\nIt follows that\n\\[ \\P \\left( \\mbox{the $I_1+I_2$ peeling steps are all nice} | \\mathcal{F}_{\\sigma} \\right) = \\frac{\\beta_{g}^{(p,p)} (\\mathbf{\\widetilde{F}}^n)}{\\beta_{g}^{(p+I_1,p+I_2)}(\\mathbf{\\widetilde{F}}^n)}. \\]\nSince $|\\mathbf{F}^n|$ is bounded by $v_0(p)$, for $n$ large enough (where \"large enough\" may depend on $p$), the Bounded ratio Lemma applies to $\\mathbf{\\widetilde{F}}^n$. Therefore, by the Bounded ratio Lemma (more precisely, by Corollary~\\ref{lem_BRL_boundaries}, item 2), the last ratio is always larger than a constant $\\eta$ depending on $\\varepsilon$. More precisely $\\eta$ may depend on $I_1$ and $I_2$, but $0 \\leq I_1, I_2 \\leq C(\\varepsilon)$, so $(I_1, I_2)$ can take finitely many values given $\\varepsilon$, so $\\eta$ only depends on $\\varepsilon$ (and not on $p$). This is the value of $\\eta$ that we choose for \\eqref{eqn_fq_approximation}. For $i \\in \\{1,2\\}$, we write $\\tau_p^{n,i}=\\sigma_{[p,p+C]}^{n,i}+I_i$. If the last $I_1+I_2$ peeling steps are nice, then after they are performed, both explored regions have perimeter $2p$. Therefore, it follows from the last computation and from \\eqref{eqn_two_distinct_regions} that, for $n$ large enough, we have\n\\[ \\P \\left( \\begin{array}{c}\\mathcal{E}^{n,1}_{\\tau_p^{n,1}} \\mbox{ and } \\mathcal{E}^{n,2}_{\\tau_p^{n,2}} \\mbox{ are both face-disjoint, have perimeter $2p$}\\\\\\mbox{and volume $\\leq v_0$, and } \\left| \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 1}_{\\sigma^{\\infty,1}_{[p,p+C]}} \\right) - \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 2}_{\\sigma^{\\infty,2}_{[p,p+C]}} \\right) \\right| \\geq \\frac{3}{4}\\varepsilon\\end{array} \\right) \\geq \\frac{4}{5} \\varepsilon \\eta.\\]\nFinally, we can use \\eqref{eqn_fq_approximation} to replace back the approximations $\\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, i}_{\\sigma^{\\infty,i}_{[p,p+C]}} \\right)$ by $r_j(\\mathbf{Q}^i)$. We obtain\n\\[ \\P \\left( \\begin{array}{c}\n\\mathcal{E}^{1,n}_{\\tau_p^{n,i}} \\mbox{ and } \\mathcal{E}^{2,n}_{\\tau_p^{n,2}} \\mbox{ are both face-disjoint, have perimeter $2p$ }\\\\ \\mbox{and $\\leq v_0$ edges, and } \\left| r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2) \\right| \\geq \\frac{1}{2}\\varepsilon\n\\end{array} \\right) \\geq \\frac{3}{5} \\varepsilon \\eta.\\]\nOn this event, we have $(M_n, e_n^1, e_n^2) \\in \\mathscr{S}_{n,p,v_0}$. Therefore, this concludes the proof of the proposition, with $\\delta=\\frac{3}{5} \\eta \\varepsilon$.\n\\end{proof}\n\n\\subsection{The two holes argument: proof of Proposition~\\ref{prop_two_holes_argument} and Corollary~\\ref{corr_main_minus_monotonicity}}\n\\label{subsec_two_holes}\n\nNow that we have Proposition~\\ref{prop_finding_regions_to_cut}, the proof of Proposition~\\ref{prop_two_holes_argument} is basically the same as two holes argument in~\\cite{BL19} (i.e. the proof of Proposition 18). Therefore, we will not write the argument in full details, but only sketch it. We first stress two differences:\n\\begin{itemize}\n\\item\nThe first one is that the involution obtained by (possibly) swapping the two explored parts is now non-identity on a relatively small set of maps (but still on a positive proportion). The only consequence is that in the end, instead of contradicting the almost sure convergence of Proposition~\\ref{prop_q_as_limit} on an event of probability $\\varepsilon$, we will contradict it on an event of probability $\\delta<\\varepsilon$, where $\\delta$ is given by Proposition~\\ref{prop_finding_regions_to_cut}.\n\\item\nThe other difference is that in \\cite{BL19}, the only observable we were using to approximate the Boltzmann weights was the ratio between perimeter and volume, which corresponds to our function $r_{\\infty}$. Here we also need to deal with the functions $r_j$ for $j \\in \\mathbb N^*$. For this, we simply need the observation that, if $q$ is much larger than $p$, the proportion of peeling steps before $\\tau_q$ where we discover a new face of perimeter $2j$ depends almost only on the part of the exploration between $\\tau_p$ and $\\tau_q$.\n\\end{itemize}\n\n\n\\begin{proof}[Sketch of proof of Proposition~\\ref{prop_two_holes_argument}]\nFix $j \\in \\mathbb N^* \\cup \\{ \\infty\\}$. Let $\\varepsilon>0$, and assume\n\\begin{equation}\\label{eqn_q1_ne_q2}\n\\P \\left( |r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2)|>\\varepsilon \\right)>\\varepsilon.\n\\end{equation}\nLet $\\delta>0$ be given by Proposition~\\ref{prop_finding_regions_to_cut}. Consider $p$ large (depending on $\\varepsilon$) and let $v_0$ be given by Proposition~\\ref{prop_finding_regions_to_cut}. We assume that the peeling algorithm $\\mathcal{A}$ that we work with has the property that the edge peeled at time $t$ for $t \\geq \\tau_p$ only depends on $\\mathcal{E}_t^{\\mathcal{A}}(m) \\backslash \\mathcal{E}_{\\tau_p}^{\\mathcal{A}}(m)$ (see~\\cite{BL19} for a more careful description of this property). This is not a problem since Proposition~\\ref{prop_finding_regions_to_cut} is independent of the choice of $\\mathcal{A}$.\n\nWe define an involution $\\Phi_n$ on the set of bi-rooted maps with genus $g_n$ and face degrees $\\mathbf{f}^n$ as follows: if $m \\in \\mathscr{S}_{n,p,v_0}$, then $\\Phi_n(m)$ is obtained from $m$ by swapping the two regions $\\mathcal{E}^{1,n}_{\\tau_p^{1,n}}$ and $\\mathcal{E}^{2,n}_{\\tau_p^{2,n}}$. If $m \\notin \\mathscr{S}_{n,p,v_0}$, then $\\Phi_n(m)=m$. Note that $\\Phi_n(M_n,e^1_n,e^2_n)$ is still uniform on bi-rooted maps with prescribed genus and face degrees. This map rooted at $e_1^n$ converges to a map $\\widehat{M}$, with either\n\\[\\widehat{M}=M^1_{\\mathbf{Q}^1}\\]\nor\n\\[\\mathcal{E}^{\\mathcal{A}}_{\\tau^1_p}(\\widehat{M}) = \\mathcal{E}^1_{\\tau^1_p} \\quad \\mbox{ and } \\quad \\widehat{M} \\backslash \\mathcal{E}^{\\mathcal{A}}_{\\tau^1_p}(\\widehat{M}) = M^2_{\\mathbf{Q}^2} \\backslash \\mathcal{E}^2_{\\tau^2_p}. \\]\n\nMoreover, by Proposition~\\ref{prop_finding_regions_to_cut}, if $p$ has been chosen large enough, then with probability at least $\\delta$, we are in the second case and furthermore $|r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2)|>\\frac{\\varepsilon}{2}$. Now assume that this last event occurs and that $q \\gg p \\gg 1$. Then we have\n\\begin{equation}\\label{eqn_rtilde_and_r_badcase}\n\\widetilde{r}_j \\left( \\mathcal{E}^{\\mathcal{A}}_{\\tau^1_p}(\\widehat{M}) \\right) \\approx r_j(\\mathbf{Q}^1) \\quad \\mbox{ and } \\quad \\widetilde{r}_j \\left( \\mathcal{E}^{\\mathcal{A}}_{\\widehat{\\tau}_q}(\\widehat{M}) \\right) \\approx r_j(\\mathbf{Q}^2),\n\\end{equation}\nwhere $\\widehat{\\tau}_q$ is the first step where the perimeter of the explored part of $\\widehat{M}$ is at least $q$. The approximations of \\eqref{eqn_rtilde_and_r_badcase} can be made arbitrarily precise if $p$ and $q$ were chosen large enough, so for $p$ large enough and $q$ large enough (depending on $p$), we have\n\\begin{equation}\\label{eqn_rj_different}\n\\P \\left( \\left| \\widetilde{r}_j \\left( \\mathcal{E}^{\\mathcal{A}}_{\\tau^1_p}(\\widehat{M}) \\right) - \\widetilde{r}_j \\left( \\mathcal{E}^{\\mathcal{A}}_{\\widehat{\\tau}_q}(\\widehat{M}) \\right) \\right| > \\frac{\\varepsilon}{4} \\right) \\geq \\delta.\n\\end{equation}\nOn the other hand $\\widehat{M}$ is a local limit of finite uniform maps, so by Theorem~\\ref{thm_main_more_general} it has to be a mixture of Boltzmann infinite planar maps. But then~\\eqref{eqn_rj_different} contradicts the almost sure convergence of Proposition~\\ref{prop_q_as_limit}, so~\\eqref{eqn_q1_ne_q2} cannot be true. Therefore, we must have $r_j(\\mathbf{Q}^1)=r_j(\\mathbf{Q}^2)$ a.s.. Since this is true for all $j \\in \\mathbb N^* \\cup \\{\\infty\\}$, by the last point of Proposition~\\ref{prop_q_as_limit}, we have $\\mathbf{Q}^1=\\mathbf{Q}^2$, which concludes the proof.\n\\end{proof}\n\nThe passage from Proposition~\\ref{prop_two_holes_argument} to Corollary~\\ref{corr_main_minus_monotonicity} does also not require any new idea compared to~\\cite{BL19}, so we do not write it down completely.\n\n\\begin{proof}[Sketch of the proof of Corollary~\\ref{corr_main_minus_monotonicity}]\nThe proof is basically the same as the end of the proof of the main theorem in \\cite{BL19}. The only difference is that we could not prove directly that $d(\\mathbf{q})$ and $\\left( a_j(\\mathbf{q}) \\right)_{j \\geq 1}$ are sufficient to characterize the weight sequence $\\mathbf{q}$, so the result that we obtain is only Corollary~\\ref{corr_main_minus_monotonicity} and not Theorem~\\ref{univ_main_thm}.\n\nMore precisely, by the Euler formula, any map with genus $g_n$ and face degrees $\\mathbf{f}^n$ has exactly $|\\mathbf{f}^n|$ edges and $|\\mathbf{f}^n|-\\sum_{j \\geq 1} f_j^n +2-2g_n$ vertices, so, by invariance of $M_n$ under uniform rerooting, we have\n\\[ \\mathbb E \\left[ \\frac{1}{\\mathrm{deg}_{M_n}(\\rho)} \\right] = \\frac{|\\mathbf{f}^n|-\\sum_{j \\geq 1} f_j^n +2-2g_n}{2|\\mathbf{f}^n|} \\xrightarrow[n \\to +\\infty]{} \\frac{1}{2} \\left( 1-2\\theta - \\sum_j \\alpha_j \\right). \\]\nBy the exact same argument as in \\cite{BL19}, we deduce from Proposition~\\ref{prop_two_holes_argument} that if $M_n \\to \\mathbb M_{\\mathbf{Q}}$, then $d(\\mathbf{Q})=\\frac{1}{2} \\left( 1-2\\theta - \\sum_j \\alpha_j \\right)$ a.s.. Similarly, by invariance under rerooting, for all $j \\geq 1$, we have\n\\[ \\P \\left( \\mbox{the root face of $M_n$ has degree $2j$} \\right) = \\frac{2j f_j^n}{2n} \\xrightarrow[n \\to +\\infty]{} j \\alpha_j. \\]\nBy the same argument as for the mean vertex degree, we obtain $a_j(\\mathbf{Q})=\\alpha_j$ a.s..\n\\end{proof}\n\n\\subsection{Monotonicity of the mean inverse degree}\n\\label{subsec_last_step}\n\nTo conclude the proof of the main theorem, given Corollary~\\ref{corr_main_minus_monotonicity}, it is enough to show that if $\\sum_{j \\geq 1} j^2 a_j(\\mathbf{q}) <+\\infty$, the weight sequence $\\mathbf{q}$ is completely determined by $\\left( a_j(\\mathbf{q}) \\right)_{j \\geq 1}$ and $d(\\mathbf{q})$. For all this subsection, we fix a sequence $(\\alpha_j)_{j \\geq 1}$ such that $\\sum_j j \\alpha_j =1$ and $\\sum_j j^2 \\alpha_j <+\\infty$ and $\\alpha_1<1$. We recall from Proposition~\\ref{prop_third_parametrization} that the weight sequences $\\mathbf{q}$ such that $a_j(\\mathbf{q})=\\alpha_j$ for all $j \\geq 1$ form a one-parameter family $\\left( \\mathbf{q}^{(\\omega)} \\right)_{\\omega \\geq 1}$ given by\n\\[ q_j^{(\\omega)}=\\frac{j \\alpha_j}{\\omega^{j-1}h_j(\\omega)} c_{\\mathbf{q}^{(\\omega)}}^{-(j-1)}, \\quad \\mbox{where} \\quad c_{\\mathbf{q}^{(\\omega)}} = \\frac{4}{1-\\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} \\frac{i \\alpha_i}{\\omega^{i-1} h_i(\\omega)}}. \\]\nTo prove Theorem~\\ref{univ_main_thm}, it is sufficient to prove the following.\n\n\\begin{prop}\\label{prop_monotonicity_deg}\nUnder the assumption $\\sum_{j \\geq 1} j^2 \\alpha_j<+\\infty$, the function $\\omega \\to d(\\mathbf{q}^{(\\omega)})$ is strictly decreasing.\n\\end{prop}\n\nSince we were not able to establish this result by a direct argument, we will prove it using Corollary~\\ref{corr_main_minus_monotonicity}.\n\n\\subsubsection{Basic properties of the mean inverse degree function}\n\nBefore moving on to the core of the argument, we start with some basic properties of the function $\\omega \\to d(\\mathbf{q}^{(\\omega)})$.\n\n\\begin{lem}\\label{lem_degreefunction_basic}\n\\begin{itemize}\n\\item[$\\bullet$]\nThe function $\\omega \\to d(\\mathbf{q}^{(\\omega)})$ is continuous on $[1,+\\infty)$ and analytic on $(1,+\\infty)$.\n\\item[$\\bullet$]\nWe have $d(\\mathbf{q}^{(\\omega)})>0$ for all $\\omega$ and $\\lim_{\\omega \\to +\\infty} d(\\mathbf{q}^{(\\omega)}) = 0$.\n\\item[$\\bullet$]\nWe have $d(\\mathbf{q}^{(\\omega)}) \\leq 1-\\sum_{j \\geq 1} \\alpha_j$ for all $\\omega \\geq 1$, with equality if and only if $\\omega=1$.\n\\end{itemize}\n\\end{lem}\n\n\\begin{proof}\nThe proof of the analyticity and continuity on $(1,+\\infty)$ is a bit long and delayed to Appendix~\\ref{subsec_analyticity}. The third item will follow from results of Angel, Hutchcroft, Nachmias and Ray~\\cite{AHNR16}. The other properties are quite easy.\n\nWe start with the continuity statement in the first item. The analyticity proved in Appendix~\\ref{subsec_analyticity} implies continuity on $(1,+\\infty)$ so it is sufficient to prove the continuity at $\\omega=1$. By the monotone convergence theorem, the function $\\omega \\to c_{\\mathbf{q}^{(\\omega)}}$ is continuous at $\\omega=1$, so $q_j^{(\\omega)}$ is continuous at $\\omega=1$ for all $j$. Therefore, for every finite map $m$ with one hole, we have\n\\[ \\P \\left( m \\subset \\mathbb M_{\\mathbf{q}^{(\\omega)}} \\right) \\xrightarrow[\\omega \\to 1]{} \\P \\left( m \\subset \\mathbb M_{\\mathbf{q}^{(1)}} \\right), \\]\nso $\\mathbb M_{\\mathbf{q}^{(\\omega)}} \\to \\mathbb M_{\\mathbf{q}^{(1)}}$ in distribution for the local topology. Since the inverse degree of the root vertex is bounded and continuous for the local topology, the function $\\omega \\to d(\\mathbf{q}^{(\\omega)})$ is continuous at $1$.\n\nWe now prove the second item: $d(\\mathbf{q}^{(\\omega)})>0$ is immediate by finiteness of vertex degrees and $d(\\mathbf{q}^{(\\omega)}) \\to 0$ is equivalent to proving $\\mathrm{deg}_{\\mathbb M_{\\mathbf{q}^{(\\omega)}}}(\\rho) \\to +\\infty$ in probability when $\\omega \\to +\\infty$. For this, we notice (see~\\eqref{eqn_omega_infinite} above) that when $\\omega \\to +\\infty$, we have $h_{i}(\\omega) \\to 1$ for all $i \\geq 1$ and\n\\[\\widetilde{\\nu}_{\\mathbf{q}^{(\\omega)}}(i) \\xrightarrow[\\omega \\to +\\infty]{} \\begin{cases}\n0 \\mbox{ if $i \\leq -1$,}\\\\\n(i+1) \\alpha_{i+1} \\mbox{ if $i \\geq 0$}.\n\\end{cases}\n\\]\nIn other words, the probability of any peeling step swallowing at least one boundary vertex goes to $0$ when $\\omega \\to +\\infty$. Therefore, if we perform a peeling exploration where we peel the edge on the right of $\\rho$ whenever it is possible, the probability to complete the exploration of the root in less than $k$ steps goes to $0$ for all $k$. It follows that the root degree goes to $+\\infty$ in probability.\n\nFinally, we move on to the third item. Since $\\mathbb M_{\\mathbf{q}}$ is stationary, if we denote by $\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}$ a map with the law of $\\mathbb M_{\\mathbf{q}}$ biased by the inverse of the root vertex degree, then $\\mathbb M_{\\mathbf{q}}$ is unimodular. A simple computation shows that $d(\\mathbf{q}) \\leq 1-\\sum_{j \\geq 1} \\alpha_j$ is equivalent to $\\mathbb E \\left[ \\kappa_{\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}}(\\rho) \\right] \\geq 0$ where, if $v$ is a vertex of a map $m$:\n\\[ \\kappa_m(v)=2\\pi-\\sum_{f} \\frac{\\mathrm{deg}(f)-2}{\\mathrm{deg}(f)}\\pi,\\]\nand the sum is over all faces that are incident to $v$ in $m$, counted with multiplicity.\nMoreover, we have equality if and only if $\\mathbb E[\\kappa_{\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}}(\\rho)] = 0$. The fact that $\\mathbb E[\\kappa_{\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}}(\\rho)] \\geq 0$ is then a consequence of \\cite[Theorem 1]{AHNR16}. Moreover, \\cite{AHNR16} shows the equivalence between 17 different definitions of hyperbolicity. In particular, we have $\\mathbb E[\\kappa_{\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}}(\\rho)] > 0$ (Definition 1 in \\cite{AHNR16}) if and only if $p_c0$ for bond percolation on $\\mathbb M_{\\mathbf{q}}$, which is equivalent to $\\mathbf{q}$ being critical (i.e. $\\omega=1$) by \\cite[Theorem 12.9]{C-StFlour}\\footnote{More precisely \\cite[Theorem 12.9]{C-StFlour} is about half-plane supercritical maps. Here is a way to extend it to full-plane maps: there is a percolation regime on the half-plane version of $\\mathbb M_{\\mathbf{q}}$ such that with positive probability, there are infinitely many infinite clusters. For topological reasons, at most two of them intersect the boundary infinitely many times. Hence, by changing the colour of finitely many edges, with positive probability there are two infinite clusters that do not touch the boundary. Since there is a coupling in which the half-plane version of $\\mathbb M_{\\mathbf{q}}$ is included in the full-plane version, we have with positive probability two disjoint infinite clusters in $\\mathbb M_{\\mathbf{q}}$ in a certain bond percolation regime, so $p_c0$ be small enough to satisfy\n\\begin{equation}\\label{eqn_choice_eps}\n\\varepsilon < \\min_{\\omega \\geq \\min(\\omega_0, \\omega_1)} \\left( d(\\mathbf{q}^{(1)}) - d(\\mathbf{q}^{(\\omega)}) \\right) \\quad \\mbox{and} \\quad \\varepsilon < \\min_{1 \\leq \\omega \\leq \\max(\\omega_0, \\omega_1)} d(\\mathbf{q}^{(\\omega)}).\n\\end{equation}\nNote that the existence of such an $\\varepsilon$ is guaranteed by the second and third items of Lemma~\\ref{lem_degreefunction_basic}. For $g \\geq 0$, we write\n\\[ k_{\\min}(g) = \\frac{2 \\sum_{j \\geq 1} \\alpha_j}{1-\\varepsilon-\\sum_{j \\geq 1} \\alpha_j} g \\quad \\mbox{and} \\quad k_{\\max}(g)=\\frac{1}{\\varepsilon} \\left( \\sum_{j \\geq 1} \\alpha_j \\right) g.\\]\nSince $\\varepsilon$ will be fixed until the end, we omit the dependence in $\\varepsilon$ in the notation.\nThese values were chosen so that a map with face degrees $\\mathbf{F}^{k_{\\min}(g)}$ and genus $g$ has average degree of order $\\frac{1}{\\varepsilon}$ (in particular such a map exists), whereas in a map with face degrees $\\mathbf{F}^{k_{\\max}(g)}$ and genus $g$ the genus is about $\\varepsilon$ times the size.\n\nWe now set, for $t \\geq 1$ and $g \\geq 0$:\n\\begin{align}\\label{eqn_defn_k0tg}\nk_0^t(g) &= \\min \\left\\{ k \\in \\left[ k_{\\min}(g), k_{\\max}(g) \\right] \\big| \\mathbb E \\left[ \\left( \\Omega^t_{k,g} \\right)^{-1} \\right] \\geq \\omega_0^{-1} \\right\\},\\\\\nk_1^t(g) &= \\min \\left\\{ k \\in \\left[ k_{\\min}(g), k_{\\max}(g) \\right] \\big| \\mathbb E \\left[ \\left( \\Omega^t_{k,g} \\right)^{-1} \\right] \\geq \\omega_1^{-1} \\right\\}.\\nonumber\n\\end{align}\nThe only reason why we use $\\omega_0^{-1}$ instead of $\\omega_0$ in the definition is to have a bounded quantity in the expectation, and pass easily from convergences in distribution to convergences of the expectation later. We first prove that this definition indeed makes sense and that the map $M_{\\mathbf{F}^{k_0^t(g)}, g}$ is well-defined with high probability.\n\n\\begin{lem}\\label{lem_k0_tg_reasonable}\nThere is $t_0$ depending only on $\\omega_0, \\omega_1$ such that for all $t \\geq t_0$:\n\\begin{enumerate}\n\\item\nfor $g$ large enough the number $k_0^t(g)$ is well defined;\n\\item\nfor $g$ large enough we have $k_0^t(g)>k_{\\min}(g)$;\n\\item\nwith probability $1-o(1)$ as $g \\to +\\infty$, we have $v \\left( \\mathbf{F}^{k_0^t(g)}, g \\right) \\geq \\frac{\\varepsilon}{2} \\left| \\mathbf{F}^{k_0^t(g)}\\right|$, and the same is true with $k_0^t(g)-1$ instead of $k_0^t(g)$.\n\\end{enumerate}\nMoreover, the same is true if we replace $k_0^t(g)$ with $k_1^t(g)$.\n\\end{lem}\n\n\\begin{proof}\nWe first recall that\n\\[v \\left( \\mathbf{F}^k, g \\right) = 2-2g+\\sum_{j \\geq 1} (j-1) F^k_j.\\]\nMoreover, by the law of large numbers and the definition of $\\mathbf{F}^k$, we have\n\\[\\frac{1}{k} \\sum_{j \\geq 1} (j-1) F^k_j \\xrightarrow[k \\to +\\infty]{a.s.} \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} (j-1) \\alpha_j \\quad \\mbox{and} \\quad \\frac{1}{k} \\left| \\mathbf{F}^k \\right| \\xrightarrow[k \\to +\\infty]{a.s.} \\frac{1}{\\sum_{i \\geq 1} \\alpha_i}. \\]\nFrom here, by the choice of $k_{\\min}(g)$, it follows easily that almost surely, for $g$ large enough, we have $v \\left( \\mathbf{F}^k, g \\right) \\geq \\frac{\\varepsilon}{2} \\left| \\mathbf{F}^k \\right|$ for all $k \\geq k_{\\min}(g)$. In particular, the third item of the Lemma will follow from the first and the second.\n\nWe now prove the first item. We need to prove that the minimum in the definition of $k_0^t(g)$ is over a nonempty set, so it is enough to prove that, if $t$ is larger than some $t_0$, we have\n\\begin{equation}\\label{eqn_omega_kmax}\n\\mathbb E \\left[ \\left( \\Omega^t_{k_{\\max}(g), g} \\right)^{-1} \\right]> \\omega_0^{-1}\n\\end{equation}\nfor $g$ large enough. We know that $v \\left( \\mathbf{F}^k, g \\right) \\geq \\frac{\\varepsilon}{2} \\left| \\mathbf{F}^k \\right|$ for $g$ large enough, so by Theorem~\\ref{thm_main_more_general} and Corollary~\\ref{corr_main_minus_monotonicity}, up to extracting a subsequence in $g$, we have the local convergence\n\\begin{equation}\\label{eqn_subseqlimit_kmax}\nM_{\\mathbf{F}^{k_{\\max}(g)}, g} \\xrightarrow[g \\to +\\infty]{(d)} \\mathbb M_{q^{(\\Omega_{\\max})}},\n\\end{equation}\nwhere $\\Omega_{\\max}$ is a random variable on $(1,+\\infty)$. On the other hand, by the law of large numbers and the definition of $k_{\\max}(g)$, we have\n\\[ \\frac{1}{g} v \\left( \\mathbf{F}^{k_{\\max}(g)}, g \\right) \\xrightarrow[g \\to +\\infty]{a.s.} -2+ \\frac{1}{\\varepsilon} \\sum_{j \\geq 1} (j-1)\\alpha_j \\quad \\mbox{and} \\quad \\frac{1}{g}\\left| \\mathbf{F}^{k_{\\max}(g)}\\right| \\xrightarrow[g \\to +\\infty]{a.s.} \\frac{1}{\\varepsilon}.\\]\nTherefore, the average degree in $M_{\\mathbf{F}^{k_{\\max}(g)}, g}$ tends to $-\\varepsilon+\\frac{1}{2} \\left( 1-\\sum_{j} \\alpha_j \\right) = d \\left( \\mathbf{q}^{(1)}\\right)-\\varepsilon$. By Corollary~\\ref{corr_main_minus_monotonicity}, we must have\n\\begin{equation}\\label{eqn_degree_omegamax}\nd \\left( \\mathbf{q}^{(\\Omega_{\\max})}\\right) = d \\left( \\mathbf{q}^{(1)}\\right)-\\epsilon\n\\end{equation}\nalmost surely. By the first inequality in our choice~\\eqref{eqn_choice_eps} of $\\varepsilon$, this implies $\\Omega_{\\max}<\\omega_0$ a.s., so $\\mathbb E \\left[ \\Omega_{\\max}^{-1} \\right] > \\omega_0^{-1}$.\n\nOn the other hand, let $t \\geq 1$. Since the explored map at time $t$ of a peeling exploration is a local function, the convergence~\\eqref{eqn_subseqlimit_kmax} implies\n\\begin{equation}\\label{eqn_convergence_omegamax_t}\n\\Omega^t_{k_{\\max}(g), g} \\xrightarrow[g \\to +\\infty]{(d)} r^{-1} \\left( \\frac{V_t^{(\\Omega_{\\max})} - 2 P_t^{(\\Omega_{\\max})}}{t} \\right).\n\\end{equation}\nMoreover, by the third item and the continuity in Lemma~\\ref{lem_degreefunction_basic}, we also get that $\\Omega_{\\max}$ is supported on a compact subset of $(1,+\\infty)$ depending only on $\\varepsilon$ (and not on the subsequence in $g$ that we are working with). Therefore, by the uniform convergence result of Lemma~\\ref{lem_unif_volume}, the right-hand side of~\\eqref{eqn_convergence_omegamax_t} converges in probability to $\\Omega_{\\max}$ as $t \\to +\\infty$, at a speed independent from the subsequence. Hence, remembering $\\mathbb E \\left[ \\Omega_{\\max}^{-1} \\right] > \\omega_0^{-1}$, there is $t_0$ depending only on $\\varepsilon$ such that for $t \\geq t_0$, we have\n\\[ \\mathbb E \\left[ r^{-1} \\left( \\frac{V_t^{(\\Omega_{\\max})} - 2 P_t^{(\\Omega_{\\max})}}{t} \\right)^{-1} \\right] > \\omega_0^{-1}. \\]\nCombined with~\\eqref{eqn_convergence_omegamax_t}, this implies $\\mathbb E \\left[ \\left( \\Omega^t_{k_{\\max}(g), g} \\right)^{-1} \\right] > \\omega_0^{-1}$ for $g$ large enough. We have proved that for $t \\geq t_0$, every subsequence in $g$ contains a subsubsequence along which~\\eqref{eqn_omega_kmax} holds for $g$ large enough, so~\\eqref{eqn_omega_kmax} holds for $g$ large enough. This proves the first item of the lemma.\n\nTo prove the second item, we need to show that there is $t'_0$ such that if $t \\geq t'_0$, then\n\\[ \\mathbb E \\left[ \\left( \\Omega^t_{k_{\\min}(g), g} \\right)^{-1} \\right] < \\omega_0^{-1} \\]\nfor $g$ large enough.\nThe proof is very similar to the proof of~\\eqref{eqn_omega_kmax}, so we do not write it in full details. The only difference is the computation of the average degree: this time, if $\\Omega_{\\min}$ plays the role of $\\Omega_{\\max}$ for the first item, using the definition of $k_{\\min}(g)$ we get $d \\left( q^{(\\Omega_{\\min})} \\right)=\\frac{\\varepsilon}{2}$ a.s.. Hence, by the second inequality of~\\eqref{eqn_choice_eps}, this implies $\\Omega_{\\min} > \\omega_0$, and the end of the proof is the same.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_monotonicity_deg}]\nLet $t$ be larger than the $t_0$ from Lemma~\\ref{lem_k0_tg_reasonable}. By the third item of Lemma~\\ref{lem_k0_tg_reasonable} and Theorem~\\ref{thm_main_more_general}, we know that $\\left( M_{\\mathbf{F}^{k_0^t(g)},g} \\right)_{g \\geq 0}$ is tight and any subsequential limit is of the form $\\mathbb M_{\\mathbf{Q}}$. Moreover by the law of large numbers, for all $j \\geq 1$ we have $\\frac{F^k_j}{\\left| \\mathbf{F}^k_j \\right|} \\to \\alpha_j$ a.s. when $k \\to +\\infty$. Therefore, by Corollary~\\ref{corr_main_minus_monotonicity}, any subsequential limit $\\mathbb M_{\\mathbf{Q}}$ must satisfy $a_j(\\mathbf{q})=\\alpha_j$ for all $j \\geq 1$, so $\\mathbf{Q}$ is of the form $\\mathbf{q}^{(\\Omega)}$ for some random $\\Omega$. Moreover, the same holds if $k_0^t(g)$ is replaced by $k_0^t(g)-1$, or $k_1^t(g)$, or $k_1^t(g)-1$. Therefore, for all $t \\geq t_0$, we can fix a subsequence $S^t$ (depending on $t$) such that when $g \\to +\\infty$ along $S^t$ the following convergences hold jointly:\n\\begin{align*}\nM_{\\mathbf{F}^{k_0^t(g)},g} & \\xrightarrow[g \\to +\\infty]{(d)} \\mathbb M_{\\mathbf{q}^{(\\Omega_0^t)}}, & M_{\\mathbf{F}^{k_1^t(g)},g} & \\xrightarrow[g \\to +\\infty]{(d)} \\mathbb M_{\\mathbf{q}^{(\\Omega_1^t)}},\\\\\nM_{\\mathbf{F}^{k_0^t(g)-1},g} & \\xrightarrow[g \\to +\\infty]{(d)} \\mathbb M_{\\mathbf{q}^{(\\Omega_0^{t,-})}}, & M_{\\mathbf{F}^{k_1^t(g)-1},g} & \\xrightarrow[g \\to +\\infty]{(d)} \\mathbb M_{\\mathbf{q}^{(\\Omega_1^{t,-})}},\\\\\n\\Omega^t_{k_0^t(g),g} & \\xrightarrow[g \\to +\\infty]{(d)} \\widetilde{\\Omega}^t_0, & \\Omega^t_{k_1^t(g),g} & \\xrightarrow[g \\to +\\infty]{(d)} \\widetilde{\\Omega}^t_1,\\\\\n\\Omega^t_{k_0^t(g)-1} & \\xrightarrow[g \\to +\\infty]{(d)} \\widetilde{\\Omega}^{t,-}_0, & \\Omega^t_{k_1^t(g)-1} & \\xrightarrow[g \\to +\\infty]{(d)} \\widetilde{\\Omega}^{t,-}_1,\\\\\n\\frac{g}{\\left| \\mathbf{F}^{k_0^t(g)} \\right|} & \\xrightarrow[g \\to +\\infty]{(d)} \\theta_0^t, & \\frac{g}{\\left| \\mathbf{F}^{k_1^t(g)} \\right|} & \\xrightarrow[g \\to +\\infty]{(d)} \\theta_1^t,\n\\end{align*}\nwhere the first four convergences are for the local topology. We highlight that for our purpose it is sufficient to consider \\emph{one} subsequence $S^t$, and that we will not try to understand \\emph{all} subsequential limits. In what follows, we will always consider $g$ in the subsequence $S^t$ and omit to precise it. Note that we have $\\Omega_0^t, \\Omega_0^{t,-} \\in [1,+\\infty)$ and $\\widetilde{\\Omega}_0^t, \\widetilde{\\Omega}_0^{t,-} \\in [1,+\\infty]$, since we have no bound a priori on $\\Omega^t_{k_0^t(g),g}$.\nWe also have $\\theta_0^t \\in \\left[ \\varepsilon, \\frac{1}{2} \\left( 1-\\varepsilon-\\sum_j \\alpha_j \\right) \\right]$ by the inequality $k_{\\min}(g) \\leq k_0^t(g) \\leq k_{\\max}(g)$ and the law of large numbers to estimate $\\mathbf{F}^{k_{\\min}(g)}$ and $\\mathbf{F}^{k_{\\max}(g)}$. By Corollary~\\ref{corr_main_minus_monotonicity}, we also know that almost surely\n\\begin{equation}\\label{eqn_omega0_in_compact}\nd \\left( \\mathbf{q}^{(\\Omega_0^t)} \\right) = d \\left( \\mathbf{q}^{(\\Omega_0^{t,-})} \\right) = \\frac{1}{2} \\left( 1-2\\theta_0^t-\\sum_{j \\geq 1} \\alpha_j \\right).\n\\end{equation}\nIn particular $d \\left( \\mathbf{q}^{(\\Omega_0^t)} \\right)$ and $d \\left( \\mathbf{q}^{(\\Omega_0^{t,-})} \\right)$ are bounded away from $0$ and $d(\\mathbf{q}^{(1)})$. By Lemma~\\ref{lem_degreefunction_basic}, this implies that $\\Omega_0^t$ and $\\Omega_0^{t,-}$ take their values in a compact subset of $(1,+\\infty)$ that depends only on $\\varepsilon$. Therefore, there is a subsequence $S$ such that, when $t \\to +\\infty$ along $S$ the following convergences hold jointly: \n\\begin{align*}\n\\Omega_0^t & \\xrightarrow[t \\to +\\infty]{(d)} \\Omega_0, & \\Omega_1^t & \\xrightarrow[t \\to +\\infty]{(d)} \\Omega_1,\\\\\n\\Omega_0^{t,-} & \\xrightarrow[t \\to +\\infty]{(d)} \\Omega_0^{-} , & \\Omega_1^{t,-} & \\xrightarrow[t \\to +\\infty]{(d)} \\Omega_1^{-},\\\\\n\\widetilde{\\Omega}_0^t & \\xrightarrow[t \\to +\\infty]{(d)} \\widetilde{\\Omega}_0, & \\widetilde{\\Omega}_1^t & \\xrightarrow[t \\to +\\infty]{(d)} \\widetilde{\\Omega}_1,\\\\\n\\widetilde{\\Omega}_0^{t,-} & \\xrightarrow[t \\to +\\infty]{(d)} \\widetilde{\\Omega}_0^{-} , & \\widetilde{\\Omega}_1^{t,-} & \\xrightarrow[t \\to +\\infty]{(d)} \\widetilde{\\Omega}_1^{-},\\\\\n\\theta_0^t & \\xrightarrow[t \\to +\\infty]{} \\theta_0, & \\theta_1^t & \\xrightarrow[t \\to +\\infty]{} \\theta_1,\n\\end{align*}\nwhere $\\Omega_0, \\Omega_0^- \\in (1,+\\infty)$ and $\\widetilde{\\Omega}_0, \\widetilde{\\Omega}_0^- \\in [1,+\\infty]$ and $\\theta_0 \\in \\left[ \\varepsilon, \\frac{1}{2} \\left( 1-\\varepsilon-\\sum_j \\alpha_j \\right) \\right]$. From now on, we will always consider $t \\geq t_0$ with $t$ in this subsequence $S$, and we will omit to precise it. \n\nThe rough sketch of the argument is to show that\n\\begin{equation}\\label{eqn_cyclic_ineq_omega}\n\\mathbb E \\left[ \\left( \\widetilde{\\Omega}^{-}_0 \\right)^{-1} \\right] \\leq \\omega_0^{-1} \\leq \\mathbb E \\left[ \\left( \\widetilde{\\Omega}_0 \\right)^{-1} \\right] = \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right] \\leq \\mathbb E \\left[ \\left( \\Omega_0^{-} \\right)^{-1} \\right] = \\mathbb E \\left[ \\left( \\widetilde{\\Omega}^{-}_0 \\right)^{-1} \\right],\n\\end{equation}\nand deduce from the equality in the third inequality that $\\Omega_0$ is deterministic and therefore equal to $\\omega_0$.\n\nMore precisely, by definition of $k_0^t(g)$ and since $k_0^t(g)>k_{\\min}(g)$, we have\n\\[ \\mathbb E \\left[ \\left( \\Omega^t_{k_0^t(g)-1} \\right)^{-1} \\right] < \\omega_0^{-1} \\leq \\mathbb E \\left[ \\left( \\Omega^t_{k_0^t(g)} \\right)^{-1} \\right].\\]\nBy letting $g \\to +\\infty$ (along $S^t$) and then $t \\to +\\infty$ (along $S$), this implies\n\\begin{equation}\\label{eqn_ineq_omegatilde}\n\\mathbb E \\left[ \\left( \\widetilde{\\Omega}^{-}_0 \\right)^{-1} \\right] \\leq \\omega_0^{-1} \\leq \\mathbb E \\left[ \\left( \\widetilde{\\Omega}_0 \\right)^{-1} \\right],\n\\end{equation}\nwhich are the first and second inequalities in~\\eqref{eqn_cyclic_ineq_omega}.\n\nOn the other hand, the third inequality will be obtained by the argument sketched in the beginning of this Subsection~\\ref{subsubsec_final_argument}. Let us first look at the relation between $M_{\\mathbf{F}^{k}, g}$ and $M_{\\mathbf{F}^{k-1}, g}$.\nWe recall that $\\mathbf{F}^{k}=\\mathbf{F}^{k-1}+\\mathbf{1}_{J_k}$, where $\\P (J_k=j)=\\frac{\\alpha_j}{\\sum_{i \\geq 1} \\alpha_i}$ for all $j \\geq 1$. If we condition on $\\mathbf{F}^{k-1}$ and $\\mathbf{F}^{k}$, then the law of $M_{\\mathbf{F}^{k-1}, g}$ is the law of $M_{\\mathbf{F}^{k}, g} \\backslash m_{J_k}^0$, conditioned on $m_{J_k}^0 \\subset M_{\\mathbf{F}^{k}, g}$, where $m^0_{J_k}$ is the map with perimeter $2$ of Figure~\\ref{fig_m_0_j}. Therefore, for any map $m$ with one hole, we have\n\\[ \\P \\left( m \\subset M_{\\mathbf{F}^{k-1}, g} | J_k=j \\right) = \\P \\left( m+m_j^0 \\subset M_{\\mathbf{F}^{k}, g} | J_k=j, \\, m_j^0 \\subset M_{\\mathbf{F}^{k}, g} \\right), \\]\nwhere $m+m_{j}^0$ is the map obtained from $m$ by replacing the root edge of $m$ by a copy of $m_j^0$. By summing over $j$, we obtain\n\\[\n\\P \\left( m \\subset M_{\\mathbf{F}^{k-1}, g} \\right) = \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{\\P \\left( m+m_j^0 \\subset M_{\\mathbf{F}^{k}, g} \\right)}{\\P \\left( m_j^0 \\subset M_{\\mathbf{F}^{k}, g} \\right)}.\n\\]\nWe now take $k=k_0^t(g)$ and let $g \\to +\\infty$ (along $S^t$) to replace $M_{\\mathbf{F}^{k-1}, g}$ and $M_{\\mathbf{F}^{k}, g}$ by respectively $\\mathbb M_{\\mathbf{q}^{(\\Omega_0^{t,-})}}$ and $\\mathbb M_{\\mathbf{q}^{(\\Omega_0^{t})}}$. We note that $m_j^0+m$ has the same perimeter as $m$ but one more internal face of degree $2j$. We obtain, for every finite map $m$ with one hole,\n\\[ \\mathbb E \\left[ C_{|\\partial m|} \\left( \\mathbf{q}^{(\\Omega_0^{t,-})} \\right) \\prod_{f \\in m} q^{(\\Omega_0^{t,-})}_{\\mathrm{deg}(f)\/2} \\right] = \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{\\mathbb E \\left[ C_{\\partial m} \\left( \\mathbf{q}^{(\\Omega_0^{t})} \\right) \\times \\prod_{f \\in m} q^{(\\Omega_0^{t})}_{\\mathrm{deg}(f)\/2} \\times q^{(\\Omega_0^{t})}_j \\right]}{ \\mathbb E \\left[ C_1 \\left( \\mathbf{q}^{(\\Omega_0^{t})} \\right) q_j^{(\\Omega_0^{t})} \\right]}. \\]\nThis can be interpreted as a Radon--Nikodym derivative, i.e. the map $\\mathbb M_{\\mathbf{q}^{(\\Omega_0^{t,-})}}$ has the law of $\\mathbb M_{\\mathbf{q}^{(\\Omega_0^{t})}}$ biased by\n\\begin{equation}\\label{eqn_radon_nikodym_omega}\n\\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{q_j^{(\\Omega_0^{t})}}{\\mathbb E \\left[ q_j^{(\\Omega_0^{t})} \\right]},\n\\end{equation}\nusing the fact that $C_1(\\mathbf{q})=1$. Since $\\Omega$ is a measurable function of the map $\\mathbb M_{\\mathbf{q}^{(\\Omega)}}$ by Proposition~\\ref{prop_q_as_limit}, it follows that $\\Omega_0^{t,-}$ has the law of $\\Omega_0^t$ biased by \\eqref{eqn_radon_nikodym_omega}. In particular, we have\n\\[\n\\mathbb E \\left[ \\left( \\Omega_0^{t,-} \\right)^{-1} \\right] = \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{\\mathbb E \\left[ q_j^{(\\Omega_0^t)} \\, \\left( \\Omega_0^t \\right)^{-1} \\right]}{\\mathbb E \\left[ q_j^{(\\Omega_0^t)}\\right]}.\n\\]\nWe can now let $t \\to +\\infty$ (along $S$) to get\n\\begin{equation}\\label{eqn_expectation_omega_inverse}\n\\mathbb E \\left[ \\left( \\Omega_0^{-} \\right)^{-1} \\right] = \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{\\mathbb E \\left[ q_j^{(\\Omega_0)} \\, \\left( \\Omega_0 \\right)^{-1} \\right]}{\\mathbb E \\left[ q_j^{(\\Omega_0)}\\right]}.\n\\end{equation}\nBut by Lemma~\\ref{lem_qj_is_monotone}, we have $\\mathbb E \\left[ q_j^{(\\Omega_0)} \\, \\left( \\Omega_0 \\right)^{-1} \\right] \\geq \\mathbb E \\left[ q_j^{(\\Omega_0)} \\right] \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right]$ for all $j \\geq 1$, so the last display implies $\\mathbb E \\left[ \\left( \\Omega_0^{-} \\right)^{-1} \\right] \\geq \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right]$, which is the third inequality of the sketch~\\eqref{eqn_cyclic_ineq_omega}.\n\nWe now move on to the two equalities of~\\eqref{eqn_cyclic_ineq_omega}. For this, we need to argue that $\\widetilde{\\Omega}_0^t$ is a good approximation of $\\Omega_0^t$ for $t$ large. By definition of $\\widetilde{\\Omega}_0^t$ and by local convergence, we have (the limits in $g$ are along $S^t$)\n\\begin{align}\\label{eqn_omegatilde_t_as_limit}\n\\mathbb E \\left[ \\left( \\widetilde{\\Omega}_0^t \\right)^{-1} \\right] &= \\lim_{g \\to +\\infty} \\mathbb E \\left[ \\left( \\Omega_{k_0^t(g),g}^t \\right)^{-1} \\right] \\nonumber \\\\\n&= \\lim_{g \\to +\\infty} \\mathbb E \\left[ \\left( r^{-1} \\left( \\frac{ \\left| \\mathcal{E}_t^{\\mathcal{A}} \\left( M_{\\mathbf{F}^{k_0^t(g)},g} \\right) \\right| - 2 \\left| \\partial \\mathcal{E}_t^{\\mathcal{A}} \\left( M_{\\mathbf{F}^{k_0^t(g)},g} \\right) \\right|}{t} \\right) \\right)^{-1} \\right] \\nonumber\n\\\\\n&= \\mathbb E \\left[ \\left( r^{-1} \\left( \\frac{V_t^{(\\Omega_0^t)}-2 P_t^{(\\Omega_0^t)}}{t} \\right) \\right)^{-1} \\right].\n\\end{align}\nWhen $t \\to +\\infty$ (along $S$), the left-hand side of~\\eqref{eqn_omegatilde_t_as_limit} goes to $\\mathbb E \\left[ \\left( \\widetilde{\\Omega}_0 \\right)^{-1} \\right]$. On the other hand, we recall that~\\eqref{eqn_omega0_in_compact} implies that $\\Omega_0^t$ lies in a compact subset of $(1,+\\infty)$ depending only on $\\varepsilon$. Since $\\Omega_0^t \\to \\Omega_0$ along $S$, by Lemma~\\ref{lem_unif_volume} we have the convergence (along $S$)\n\\[ \\frac{V_t^{(\\Omega_0^t)} -2P_t^{(\\Omega_0^t)}}{t} \\xrightarrow[t \\to +\\infty]{(P)} r(\\Omega_0). \\]\nTherefore, when $t \\to +\\infty$ (along $S$), the right-hand side of~\\eqref{eqn_omegatilde_t_as_limit} goes to $\\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right]$. This proves the first equality of~\\eqref{eqn_cyclic_ineq_omega}. The second one is proved in the exact same way, using $\\Omega^{t,-}_0$, $\\widetilde{\\Omega}^{t,-}_0$ instead of $\\Omega^{t}_0$, $\\widetilde{\\Omega}^{t}_0$.\n\nWe have therefore proved all of~\\eqref{eqn_cyclic_ineq_omega}, so all the inequalities must be equalities. In particular, \\eqref{eqn_expectation_omega_inverse} becomes\n\\[ \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right] = \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{\\mathbb E \\left[ q_j^{(\\Omega_0)} \\, \\left( \\Omega_0 \\right)^{-1} \\right]}{\\mathbb E \\left[ q_j^{(\\Omega_0)}\\right]}. \\]\nHowever, we also know by Lemma~\\ref{lem_qj_is_monotone} that $\\mathbb E \\left[ q_j^{(\\Omega_0)} \\, \\left( \\Omega_0 \\right)^{-1} \\right] \\geq \\mathbb E \\left[ q_j^{(\\Omega_0)} \\right] \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right]$ for all $j$, so for all $j \\geq 1$ we must have the equality\n\\[ \\alpha_j \\mathbb E \\left[ q_j^{(\\Omega_0)} \\left( \\Omega_0 \\right)^{-1} \\right] = \\alpha_j \\mathbb E \\left[ q_j^{(\\Omega_0)} \\right] \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right].\\]\nIn particular, we fix $j \\geq 2$ such that $\\alpha_j>0$ (such a $j$ exists because $\\alpha_1<1$). Then $\\mathbb E \\left[ q_j^{(\\Omega_0)} \\left( \\Omega_0 \\right)^{-1} \\right] = \\mathbb E \\left[ q_j^{(\\Omega_0)} \\right] \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right]$. Since $\\omega \\to \\omega^{-1}$ and $\\omega \\to q_j^{(\\omega)}$ are two decreasing functions (by Lemma~\\ref{lem_qj_is_monotone}), this is only possible if $\\Omega_0$ is deterministic. But then~\\eqref{eqn_cyclic_ineq_omega} yields $\\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right] = \\omega_0^{-1}$, so $\\Omega_0=\\omega_0$ a.s..\n\nWe can now finish the proof. By the exact same argument as for $\\Omega_0$, we also have $\\Omega_1=\\omega_1$ a.s.. We recall that $\\omega_0<\\omega_1$. By letting $t \\to +\\infty$ (along $S$) in~\\eqref{eqn_omega0_in_compact} and using the continuity result of Lemma~\\ref{lem_degreefunction_basic}, we get\n\\begin{equation}\\label{eqn_fin_omega0_theta0}\nd \\left( \\mathbf{q}^{(\\omega_0)} \\right) = \\frac{1}{2} \\left( 1-2\\theta_0-\\sum_{j \\geq 1} \\alpha_j \\right)\n\\end{equation}\nand similarly\n\\begin{equation}\\label{eqn_fin_omega1_theta1}\nd \\left( \\mathbf{q}^{(\\omega_1)} \\right) = \\frac{1}{2} \\left( 1-2\\theta_1-\\sum_{j \\geq 1} \\alpha_j \\right).\n\\end{equation}\nOn the other hand, by the definition~\\eqref{eqn_defn_k0tg} of $k_0^t(g)$ and $k_1^t(g)$, since $\\omega_0<\\omega_1$, we have\n\\[k_0^t(g) \\geq k_1^t(g)\\]\nfor all $t$ and $g$. Therefore, we have $\\left| \\mathbf{F}^{k_0^t(g)} \\right| \\geq \\left| \\mathbf{F}^{k_1^t(g)} \\right|$. By letting $g \\to +\\infty$ (along $S^t$) we deduce $\\theta_0^t \\leq \\theta_1^t$. Letting $t \\to +\\infty$ (along $S$) we get $\\theta_0 \\leq \\theta_1$. Combining this with~\\eqref{eqn_fin_omega0_theta0} and~\\eqref{eqn_fin_omega1_theta1}, this proves $d \\left( \\mathbf{q}^{(\\omega_0)} \\right) \\geq d \\left( \\mathbf{q}^{(\\omega_1)} \\right)$, so the function $\\omega \\to d \\left( \\mathbf{q}^{(\\omega)} \\right)$ is nonincreasing on $(1,+\\infty)$. Since it is nonconstant (for example by the second item of Lemma~\\ref{lem_degreefunction_basic}) and analytic (first item of Lemma~\\ref{lem_degreefunction_basic}), it is decreasing on $(1,+\\infty)$. Finally, we extend the result to $[1,+\\infty)$ by continuity at $1$ (first item of Lemma~\\ref{lem_degreefunction_basic}).\n\\end{proof}\n\n\\section{Asymptotic enumeration: convergence of the ratio}\\label{sec_univ_asymp}\n\n\\begin{proof}[Proof of Corollary \\ref{prop_cv_ratio}]\nFix $j \\geq 1$, and let $m_j^0$ be the map of Figure~\\ref{fig_m_0_j} with a hole of perimeter $2$. On the one hand, we have\n\\[\\P \\left( m_j^0 \\subset \\mathbb{M}_{\\mathbf{q}} \\right)=C_2(\\mathbf{q})q_j.\\]\nOn the other hand, \n\\[\\P \\left( m_j^0 \\subset M_{\\mathbf{f}^{n},g_n} \\right)=\\frac{\\beta_{g_n}(\\mathbf{f}^{n}-\\mathbf{1}_j)}{\\beta_{g_n}(\\mathbf{f}^{n})}.\\]\nThe last equality is proved by contracting $m_j^0$ in $M_{\\mathbf{f}^{n},g_n}$ into the root edge of a map with face degrees given by $\\mathbf{f}^{n}-\\mathbf{1}_j$. The corollary follows by letting $n \\to +\\infty$.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}