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+{"text":"\\section{Introduction}\nICRU Report 36 on microdosimetry~[\\onlinecite{ICRU36}] provides general formulations for the mean chord length in terms of the geometrical dimension of the sensitive volumes of interest (SVOI).\nAccordingly, $\\langle l \\rangle \\equiv \\overline{l}$ is the mean length of randomly oriented chords in that volume and can be calculated based on the geometrical structure of the target of interest.\nThe mean chord length that is an intrinsic geometrical character of SVOI, results from the random interception of the SVOI by a straight geometrical line, superseded for a (single) physical track of a charged particle.\nThe chord length calculated as such is the mean of shortest Euclidean distances between pairs of end points, crossing closed surface of SVOI.\n\nAs originally introduced by Rossi, Kellerer and colleagues (e.g., see for example Refs.~[\\onlinecite{Kellerer1971:RR,Kellerer_Rossi1972:CTRR,Kellerer1984:RR,Kellerer1985:Book,Kellerer1975:REB}]), the lineal energy, $y = \\varepsilon \/ \\langle l \\rangle$,\nis the energy imparted in a single event, $\\varepsilon$, divided by the mean chord length, $\\langle l \\rangle$.\n$y$ is a random variable analogous to particle linear energy transfer (LET). Accordingly, the randomness stems from $\\varepsilon$ as the chord-length has been averaged out and it is a predetermined number.\nFor a convex volume, $V$, such as spherical geometries with area $A$, a pure geometrical calculation, first performed by Cauchy~[\\onlinecite{Cauchy1908}], yields $\\langle l \\rangle = 4V\/A$.\nFor a rectangular slab with the transversal thickness $d$ perpendicular to the unidirectional beam central axis, $\\langle l \\rangle = d$~[\\onlinecite{Magrin2018:PMB}].\nThe mean chord length, as such, which is based on straight tracks in a micro-meter size in a SVOI, has been vastly used in microdosimetry literature as well as in modelings in the radio-biological effects of any field of radiation.\n\nOne of the first studies based on MC technique to calculate chord-length distributions was reported by Birkhoff {\\em et al.}~[\\onlinecite{Birkhoff1970:HP}].\nIn this type of MC calculations, particles are not scored via a collision-by-collision approach using coarse-grained quantum mechanical scattering cross-sections, as we perform in now-a-days MC models such as Geant4~[\\onlinecite{Agostinelli2003:NIMA,Allison2006:TNS,Allison2016:NIMPRA}] and Geant4-DNA~[\\onlinecite{Incerti2010:IJMSSC}] where the tracks, including primary and secondary particles, exhibit the inchoate distribution of energy transfers~[\\onlinecite{Conte2012:NJP,Abolfath2011:JPC,Abolfath2013:PMB,Abolfath2016:MP,Abolfath2017:SR,Abolfath2019:EPJD}].\nInstead, Birkhoff {\\em et al.}~[\\onlinecite{Birkhoff1970:HP}] considered straight geometrical lines intercepting a sensitive volume of energy-proportional devices such as spherical \/ cylindrical gas proportional counters.\nThe rationale behind these simplifications is based on uniform and continuous energy transfer to matter along the tracks of charged particles.\nMoreover, the tracks considered in simulations performed by Birkhoff {\\em et al.}, were assumed to be part of a uniform source beam of radiation.\n\nThe MC techniques as such,\nare therefore analogous to the classical Metropolis MC algorithm proposed for calculation of the numerical value of $\\pi$, or numerical multi-variable integrations~[\\onlinecite{NumericalRecipes}].\nBy taking a large random sample, the resulting chord-length distribution approaches the correct geometrical distribution in a statistical manner.\n\n\nThis is seemingly a plausible approach, applicable to cavities of the instruments designed for measuring pulse height distributions in a field of radiation such as energy-proportional devices, as details in localities of the collisions, their spatial distribution and compactness within the cavity sensitive volume are irrelevant to overall observable electrical responses. The mathematical model including the calculation of the chord length,\nas presented in Birkhoff {\\em et al.}~[\\onlinecite{Birkhoff1970:HP}] and recent modifications and refinements in Refs.~[\\onlinecite{Bolst2017:PMB,Anderson2017:MP,Bolst2018:PMB}], attempted to introduce an alternative approach to the Cauchy path length, is consistent with the occurrence of the physical processes in micro-dosimetry devices, considering Rossi's formulation of lineal energy $y$.\n\nThe responses from biological systems, however, seems slightly different from the physical responses we expect to observe from the design of micro-dosimetry proportional counters.\nFor example, the nano-meter scale linear density of DNA double strand breaks (DSBs) induced by traversing of a track of charged particle in a cell nucleus that determines complexities in lethal pathways of the lesions, cell death and tissue late effects, is expected to be more sensitive to the compactness and linear distribution of the individual collisions, rather than the geometrical dimensions of the cell nuclei~[\\onlinecite{Abolfath2019:EPJD}].\nThe former roots in nano-meter scale localities in physio-chemical phenomenon and effects occurring in DNA material in cell nuclei whereas the latter is global characters associated with the outer structure of the cell nuclei boundaries and their membranes, prone to macroscopic-scale volume effects.\nMore precisely, such model calculations, as also pointed out, e.g., in Ref.~[\\onlinecite{Carlson2008:RR}], only provide an estimate of overall imparted energy within cell nucleus volume without offering an appropriate analysis in resolving nano-scale biologically relevant events, as the geometric chord length of a cell nucleus is in $\\mu m$-scale, three orders of magnitude larger than nano-meter scale, the microscopic resolution of DNA-damage.\n\nMoreover, as particles penetrate in tissue, they lose their kinetic energy, hence the compactness of collisions and the complexities in DNA damage are expected to rise. The geometrical chord-length as calculated by Cauchy and implemented in microdosimetry, is a fixed number, e.g., a factor proportional to the dimension of cell nucleus and does not vary as a function of depth.\n\nLet us demonstrate the subtleties and lack of sufficiency in the standard microdosimetry formalism to fully describe and capturing the microscopic \/ nano-meter responses and critical variabilities in biological complexities,\nby considering passage of a single charged particle in a $\\mu$m-scale SVOI, a typical representation of a cell nucleus, as schematically shown in Fig.~\\ref{fig0}.\nHence occurrence of various damages in DNA-materials and chromosomes such as double strand breaks (DSBs) are expected to be seen in these volumes~[\\onlinecite{Abolfath2011:JPC,Abolfath2013:PMB,Bianco2015:RRD}].\n\nIn Fig.~\\ref{fig0}, the black cross-lines ($\\times$) represent locations of the site of damages in DNA materials connected diagrammatically to the charged particle by the wiggly lines. The lines are Feynman diagram representation of the photon field propagators in quantum electrodynamics (QED) that describe interaction of charged particles in scattering processes with random interaction sites on DNA and\/or the environment of DNA in a cell nucleus.\nA process that describes release of OH-free radicals and\/or reactive oxygen species (ROS) in indirect DNA damage (the latter), or shell electrons localized initially in DNA in direct damage processes (the former).\nFor further details in simulating such microscopic events at nano-scales, we refer the readers to our previous studies and publications, for example Ref.~[\\onlinecite{Abolfath2013:PMB}].\n\nFor the present discussion, it is critical to recognize that the locations of the sites of interaction are random, hence the path lengths and distances between two sequential interaction sites along the beam central axis, $l$, are random variable. This is in addition to randomness in energy imparted at the location of interaction, $\\varepsilon$.\n\nAlthough it should be obvious that both $l$ and $\\varepsilon$ are two random independent variables, following two independent distribution functions, in the standard microdosimetry formalism, only $\\varepsilon$ was treated directly a random variable associated with the beam quality.\nThe sub-micrometer randomness in steps in $l$ was neglected, because such information was not necessary to be recorded from the measurement theory standpoints.\nInstead a constant chord-length that is a characteristic \/ geometrical length of microdosimetric sites was superseded.\n\nMore rigourously, the statistical fluctuations in $l$ with regards to all collisions scored at nano-scale, crucial to DNA-damage statistical analysis, cannot be captured by micrometer-scale chord-length calculated by averaging $l$ over the track-distribution functions.\nIn fact, in the microdosimetry distribution functions, the location of the collisions over a single track were already traced out \/ integrated over thus any information regarding to individual collision was wiped out.\nIn our approach, presented in this work, we substituted the standard microdosimetry distribution functions by collision distribution functions.\nBecause a single track consists of several collisions in a microdosimetry site, the collision-based distribution functions of $\\varepsilon$ and $l$ are more informative and appropriate for modeling biological responses.\n\n\nIn addition to aforementioned shortcomings, substantial number of studies and publications, including the microdosimetric kinetic model (MKM)~[\\onlinecite{Hawkins1998:MP,Hawkins2003:RR,Kase2006}], implemented in the Particle and Heavy Ion Transport Code System\n(PHITS)~[\\onlinecite{Takada2018:JRR}], formulated based on an alternative approach by introducing a concept of virtual spherical domains in cell nuclei with a size that varies within nano- to $\\mu$-meter.\nIn these models, it is not clear the rational for a choice of domain size but considering it spuriously an additional phenomenological parameter.\nTherefore the application of current micro-dosimetry formulation in radio-biology does not seem appropriate and is prone to mixing the above discernible aspects.\n\nWe note that the Cauchy formula for the mean chord length is valid only for isotropic radiation (for non-spherical detectors)~[\\onlinecite{Magrin2018:PMB}].\nHadron therapy beams are, at the contrary, non-isotropic and the primary particles have essentially a unique direction.\nThus several microdosimetry publications were developed based on non-Cauchy algorithms for the mean chord length.\nHowever, the extension of Cauchy to non-Cauchy calculation of chord length does not seem to overcome the shortcomings of application of chord length in biological models, because the chord-length, by definition, is a character of a surface boundary of microdosimetry sites, whether it is a detector, cell nucleus or MKM domains.\nThe chord-length as such does not contain detailed information on occurrence of the biological events and complexities inside the bulk of the site.\n\nThus it is appealing to revisit and refine these well established micro-dosimetry models and generalize them for modelings in biological response theories, radio-biological applications, and nano-dosimetry.\nThis is the main goal of the present study to propose necessary modifications essential in tweaking micro-dosimetry and suit it for nano-dosimetry formulations.\nAlong this line of thoughts, we propose substitution of the cellular geometrical chord-length (regardless if it was computed using Cauchy or non-Cauchy techniques) by the tracks {\\it diffusive mean free path}, $\\langle\\langle l \\rangle\\rangle$.\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe demonstrate that to simulate biologically relevant passage of a charge particle, it is necessary to collect the events, collision-by-collision, and combine them together to shape a track-structure, a bottom-up approach.\nThis led us to perform a double averaging over collisions and tracks to calculate the track mean-free path, denoted by $\\langle\\langle l \\rangle\\rangle$.\nSimilar to a gas phase of matter, the molecules collide with one another, hence the diffusive mean free path is the average distance a particle travels between collisions. The larger the particles or the denser the gas, the more frequent the collisions are and the shorter the mean free path.\nTurning back to original problem, the shorter the mean free path of the charged particle tracks manifests in higher complexities in DNA damage, hence the higher biological impact.\nWe thus supersede the track-averaged chord length, $\\langle l \\rangle$, by the mean-free path, $\\langle\\langle l \\rangle\\rangle$.\nIn calculation of lineal energy and LET, to be consistent with changes in calculating $\\langle\\langle l \\rangle\\rangle$, we also supersede the single event energy imparted, $\\varepsilon$, by energy imparted in a single collision in the cellular SVOI. We then average over individual collisions and tracks in SVOI, wherever is necessary.\n\nWe substantiate our proposal by presenting numerical illustrations on recently reported experimental data based on in-{\\it vitro} clonogenic cell survival assay of non-small cell lung cancer (NSCLC) cells, i.e., an observation which was obtained by performing a high-throughput and high accuracy clonogenic cell-survival data acquired under exposure of the therapeutic scanned proton beams~[\\onlinecite{Guan2015:SR}].\nWe compare two RBE models using (1) ICRU 36 geometrical chord length of typical spherical cell nuclei with radius of $\\langle l \\rangle = 5 \\mu m$ and (2) charge particle diffusive mean-free path, $\\langle\\langle l \\rangle\\rangle$ that is a variable and a function of depth.\nWe show that a monotonic decrease in $\\langle\\langle l \\rangle\\rangle$ as a function of depth leads to a monotonic increase up to a factor of 4 in RBE.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{Fig1_epsilon_length_New.pdf}\\\\\n\\noindent\n\\caption{\nSchematically shown passage of a charged particle (red bold arrow) through a $\\mu$m-scale SVOI.\nCollision-by-collision following by event-by-event length and imparted energy scales, $\\langle\\langle l \\rangle\\rangle$ and $\\langle\\langle \\varepsilon \\rangle\\rangle$, respectively, describe typical distance between sequential interaction sites and imparted energy between charged particle and biological materials.\n}\n\\label{fig0}\n\\end{center}\\vspace{-0.5cm}\n\\end{figure}\n\n\n\\section{Method and Materials}\n\\subsection{Mathematical formalism}\nIn our recent study~[\\onlinecite{Abolfath2019:EPJD}], we demonstrated that the linear density of DNA DSBs, $\\Delta_l$,\na fundamental biological response function in a radiation field, is a variable of linear density of the collisions.\nTo this ends, we recall the following relation derived in our first principle multi-scale study~[\\onlinecite{Abolfath2019:EPJD}]\n\\begin{eqnarray}\n\\Delta_l = \\frac{\\mu}{m}\n\\frac{\\langle\\langle \\varepsilon^2 \\rangle\\rangle}{\\langle\\langle \\varepsilon \\rangle\\rangle}\n\\frac{1}{\\langle\\langle l \\rangle\\rangle},\n\\label{eq1}\n\\end{eqnarray}\nwhere $\\varepsilon$ and $l$ are single collision energy imparted and stepping-length, calculated by Geant4 MC toolkit~[\\onlinecite{Agostinelli2003:NIMA,Allison2006:TNS,Allison2016:NIMPRA}].\nHere $\\mu$ and $m$ are the average number of DSBs per deposition of 1 Gy of ionizing dose and cellular \/ DNA mass, respectively.\n\nHence, the relevant quantity in unit of lineal-energy (energy per length), consistent with Eq.(\\ref{eq1}), was defined\n\\begin{eqnarray}\ny_{1D} = \\frac{\\langle\\langle \\varepsilon^2 \\rangle\\rangle}{\\langle\\langle \\varepsilon \\rangle\\rangle}\n\\frac{1}{\\langle\\langle l \\rangle\\rangle},\n\\label{eq2}\n\\end{eqnarray}\nsuch that\n\\begin{eqnarray}\n\\Delta_l = \\frac{\\mu}{m} y_{1D}.\n\\label{eq3}\n\\end{eqnarray}\n\nIn terms of these quantities, calculation of cell survival was performed\n\\begin{eqnarray}\n-\\ln(SF) = \\sum_{i=1}^N \\sum_{j=1}^i b_{i,j} \\Delta_l^{i-j} D^j.\n\\label{eq4}\n\\end{eqnarray}\nExpanding Eq.(\\ref{eq4}) and keeping the series up to quadratic term in dose, $D$, we found radio-biological $\\alpha$, $\\beta$ indices, as following\n\\begin{eqnarray}\n\\alpha = \\sum_{i=1}^N b_{i,1} \\Delta_l^{i-1},\n\\label{eq5_1}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\beta = \\sum_{i=1}^{N-1} b_{i+1,2} \\Delta_l^{i-1},\n\\label{eq5_2}\n\\end{eqnarray}\nThe dependence of $\\alpha$ and $\\beta$ on $y_{1D}$ can be derived from the dependence of $\\Delta_l$ on $y_{1D}$.\nThe coefficients in Eq.(\\ref{eq4}) are identified by fitting to the experimental cell-survival data.\n\n\\subsection{Geant4 Monte Carlo simulations}\nIn Geant4 each pencil proton beams were simulated by irradiating a cylinder water phantom with 20 cm radius and 20 cm length. The mean deposited energies $\\langle\\langle\\varepsilon\\rangle\\rangle$ and $\\langle\\langle\\varepsilon^2\\rangle\\rangle$, the mean track length $\\langle\\langle l \\rangle\\rangle$ were scored within a linear array of voxels with 0.5 $mm$ down to 5 $\\mu m$ thickness.\nIn our notations, averaging over all single point collisions, following by averaging over all single primary and secondary particle tracks are denoted by double averaging, $\\langle\\langle\\cdots\\rangle\\rangle$.\nIn MC we scored $y_{1D}$, $y_{D}$ and LET$_d$ from collision-by-collision following by event-by-event averaging and energy deposition $\\varepsilon_j$ and stepping length $l_j$, using the following identities\n$y_{1D} = (\\sum_j \\varepsilon_j^2\/\\sum_j \\varepsilon_j)\/(\\sum_j l_j\/\\sum_j 1_j)$, $y_D = \\sum_j (\\varepsilon_j\/l_j)^2\/\\sum_j (\\varepsilon_j\/l_j)$, and\n${\\rm LET}_d = \\sum_j (\\varepsilon_j^2\/l_j) \/ \\sum_j \\varepsilon_j$\nwhere sum over $j$ includes all energy deposition events from primary and secondary processes in all steps and tracks in a specific voxel, hence $\\sum_j 1_j$ represents total number of single-collisions, scored in all energy imparted events.\n\nTherefore $y_{1D}=(\\langle\\langle\\varepsilon^2\\rangle\\rangle\/\\langle\\langle\\varepsilon\\rangle\\rangle)\/\\langle\\langle l \\rangle\\rangle$ as well as other types of LET's were calculated.\nThe number of primary protons and the number of interactions per track were saved, in the same volume.  Then, the energy deposition, the track length, the number of primary proton and the number of interactions were accumulated, in each cell. All simulations used $10^6$ protons with series of energy cut-off, corresponding to particle range that vary within 1 $mm$ and 1 $\\mu m$.\nIn Geant4, any particle with energy below the cut-off value is assumed to not produce secondary particles (i.e. production threshold).\nBelow these cuts, the particle is transported further according to the CSDA approximation which will still imply a varying energy loss, i.e., no tracking cuts ~[\\onlinecite{Agostinelli2003:NIMA}].\nAll simulation results presented used the QGSP-BIC-EMY physics list.\nWe used Gaussian proton energy spectrums with very small FWHM (0.18 MeV). Because of small divergence the simulated beam is mono-energetic.\n\n\n\nIn Geant4, when primary particle (in our case proton) collide with other particle, secondary particles are generated.\nThis includes photon, electron, proton, neutron, He and heavier ions. The primary particle will not be processed until all secondaries are dealt with.\nThe importance of the energy cut-off, requires the program to stop producing more secondaries when the energy of the secondaries become lower than energy cut-off, defined by the user as a cut-off in length-scale.\nOtherwise the simulation will not be practical as will take considerably long time to simulate the events.\nHowever, the very low energy secondaries will not be killed, but it will follow the continuous slowing down approximation, CSDA.\nThe CSDA works fine with low energy particles as they tend to not travel far in the material.\nIn other words, the cut-off represents the accuracy of the stopping position, and any particle will always be tracked down to zero kinetic energy.\nIn our previous studies, we did validate the Geant4 simulation using the cut-off values (as we used in current study) and experimentally measured Bragg peak via production of Cerenkov light~[\\onlinecite{Helo2014:PMB}].\nThe difference was found to be less than 1\\% for cut-off of 0.01 mm.\n\nIn the current manuscript, we used the terms cut-off, and cut-all interchangeably.\nThe term cut-all used in the legend of figures to emphasize a specific cut-off value applied uniformly for ``all\" primary and secondary particles.\nNote that the corresponding cut-off values in energy depends on the type of secondary particles.\nFor example the length-scale cut-off of 0.01 mm, approximately corresponds to 0.025 MeV cut-off for electrons in water.\nThus this value of energy for proton and the rest of particles is different than the corresponding value of cut-off in energy for electron.\n\nFinally we note that the mean free path calculated by Geant4 methodology, as described above, should not be confused with the distance between individual ionizations, as this would be a true track structure approach. But from the Geant4 MC simulations, it must be clear what we calculated is a condensed history simulation where a large number of ionizations are grouped in a single energy loss step so it is rather the mean free path between individual energy loss events above a certain production cut.\n\n\n\n\n\n\\section{Results}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{Fig2_stepping_length_v0.pdf}\\\\\n\\noindent\n\\caption{\nShown $\\langle\\langle l \\rangle\\rangle$ calculated for various cutoff values as a function depth for\na pencil beam of proton with nominal energy 80 MeV, traversing a cylindrical water phantom with slice thickness equal to 0.5 $mm$.\n}\n\\label{fig1}\n\\end{center}\\vspace{-0.5cm}\n\\end{figure}\n\nIn Fig.~\\ref{fig1}, we illustrate the effect of particle cutoff length on $\\langle\\langle l \\rangle\\rangle$.\nThe calculation was performed for a pencil beam of proton with nominal energy 80 MeV, hitting a cylindrical water with radius and length of 20 cm, and slice thickness of 0.5 $mm$.\nAs shown in Fig.~\\ref{fig1}, the depth dependence of $\\langle\\langle l \\rangle\\rangle$ is strongly influenced by the user defined cutoff choice.\nThe cutoff larger or equal to size of slice thickness, as shown by the black circles and red triangles respectively, yield the collision averaged chord-length equal to the slice thickness, a result that coincides with the geometrical chord-length of the water slice-thickness.\nIn this limit of large cut-off's, our numerical results recover the CSDA limits and the geometrical chord length calculated by Cauchy, Kellerer {\\em et al.} and ICRU 36.\nWith lowering the cutoff down to 10th of slice thickness (green triangles) or even further (blue squares), the track mean-free path becomes significantly smaller than the geometrical slice thickness.\nBy lowering cutoff, below 0.01 mm, we observe negligible change in $\\langle\\langle l \\rangle\\rangle$,\ni.e., if we lower the cut-off below 0.01 mm, the generated curve exhibits slight difference relative to the curve corresponding to 0.01 mm cut-off, shown by squares in Fig.~\\ref{fig1}.\nNumerically, we reached to a domain that resulted in convergence of the output data with respect to variations in cutoff values.\nA continuous decrease in $\\langle\\langle l \\rangle\\rangle$ as a function of depth reveals an increase in linear compactness of collisions.\n\nNote that, for slice thickness of 0.5 mm, as in Fig.~\\ref{fig1}, and large cut-off values, no secondary particles are generated.\nThe primary particle, i.e., proton, loss energy according to CSDA and in results the simulated chord length turns out to be identical to the slice thickness, e.g., 0.5 mm as in Fig.~\\ref{fig1}, which fits very well with Rossi-Kellerer's theory of chord-length.\nHowever, if we lower the cut-off value the simulated chord length will become considerably less (0.15 mm as seen in Fig.~\\ref{fig1}).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{Fig3_stepping_length_PDD_v2.pdf}\\\\\n\\noindent\n\\caption{\nShown $\\langle\\langle l \\rangle\\rangle$ (black lines) and relative energy deposition (red lines) calculated for 80 MeV pencil beam of proton, traversing a cylindrical water phantom with 5.0 $\\mu m$ slice thickness, using a cutoff values for all particles equal to 1 $\\mu m$. The vertical dash line indicates the position of Bragg peak.\n}\n\\label{fig2}\n\\end{center}\\vspace{-0.5cm}\n\\end{figure}\n\nIn Fig.~\\ref{fig2}, we show $\\langle\\langle l \\rangle\\rangle$ (black lines) and relative energy deposition, $\\langle\\langle \\varepsilon \\rangle\\rangle$, (red lines) calculated for a pencil beam of $10^6$ protons with nominal energy 80 MeV, traversing a cylindrical water phantom with slice thickness equal to 5.0 $\\mu m$.\nTo score such fine slices, we divided the water phantom thickness, equal to 20 cm, into 40,000 divisions and scored collision-by-collision energy deposition, $\\varepsilon$, and stepping length, $l$, in each slice, using a cutoff values equal to 1 $\\mu m$.\nThe vertical dash line indicates the position of Bragg peak at approximate 5.2 cm depth corresponding with $LET_d \\approx 10 keV\/\\mu m$.\n\nNote that because of 5 $\\mu$m slice thickness and use of 1 $\\mu$m cut-off, the effect of delta-rays and secondary particles, imparting their energies outside of the slices they were generated, effectively alter the shape of PDD. However, the location of the Bragg peak does not change as the slice thickness reduces from 1 mm to 5 $\\mu$m.\nWe should also remark that if we lower the thickness of the SVOI to nano-scales, we must lower the cut-off values to below 1 nm to obtain sensible results.\n\n\nIt is intriguing to note that from beam entrance to the end of the proton range, $\\langle\\langle l \\rangle\\rangle$ decreases by a factor of eight.\nCoincidentally, a factor of 4 has been reported experimentally in increase in proton RBE distal and proximal to Bragg peak (for more discussion see Fig.~\\ref{fig4}, below).\nThe wiggling lines in the vicinity of proton range stem from statistical noises due to particle energy straggling and lack of enough statistics because of small number of particle fluence.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{Fig4_LETd_y1_v2.pdf}\\\\\n\\noindent\n\\caption{\nShown $y_{1D}$ vs. $LET_d$ scored for a MC setup identical to Fig.~\\ref{fig2}.\n}\n\\label{fig3}\n\\end{center}\\vspace{-0.5cm}\n\\end{figure}\n\nIn Fig.~\\ref{fig3}, $y_{1D} = \\left[\\langle\\langle \\varepsilon^2 \\rangle\\rangle \/ \\langle\\langle \\varepsilon \\rangle\\rangle \\right] \/ \\langle\\langle l \\rangle\\rangle$ vs. $LET_d = \\langle\\langle \\varepsilon^2\/l \\rangle\\rangle \/ \\langle\\langle \\varepsilon \\rangle\\rangle$ is shown.\nThe trace shown by black dots obtained from variable $\\langle\\langle l \\rangle\\rangle$, as shown in Fig.~\\ref{fig2} whereas the trace by the red dots uses an ICRU 36 type of chord-length estimate, i.e., $\\langle\\langle l \\rangle\\rangle = 5 \\mu m$ everywhere, regardless of the depth.\nNote that $\\langle\\langle l \\rangle\\rangle = 5 \\mu m$ is a geometrical chord-length for a slab with thickness equal to $5 \\mu m$.\nThe green solid line is a CSDA line, implying $LET_d = y_{1D}$ under assumption $\\langle\\langle 1\/l \\rangle\\rangle \\approx 1 \/ \\langle\\langle l \\rangle\\rangle$.\nA similar non-linear dependencies between different types of LET's were previously presented in~[\\onlinecite{Abolfath2019:EPJD}].\nA non-linear dependence between LET$_d$ and $y_{1D}$ suggests advantageous in using $y_{1D}$ over LET$_d$ for RBE studies because of linear dependence between $y_{1D}$ and $\\Delta_l$, as is given by Eq.(\\ref{eq1}).\n\nWe note that the scattered dots in Fig.~\\ref{fig3} are due to the statistical fluctuations in a MC simulation performed under limited number of primary protons. By increasing number of protons, the scattered dots merge together and form a single curve for each value of cutoff as shown in Fig.~\\ref{fig3}.\nIn this calculation we used 10$^6$ protons. Because of small thickness of slices and small cut-off value, the MC calculation is very slow. To generate similar curves with lower statistical variations, a simulation based on larger number of protons, e.g., 10$^7$ protons is required. However, because in Fig.~\\ref{fig3}, we only intend to show a non-linear dependence between LET$_d$ and y$_{1D}$, the envelope dependence obtained from scoring a million of protons is adequate for our current qualitative analysis. We note that, the red and black envelop-curves in Fig.~\\ref{fig3} were collected through merging the scattered points. We also note that we did not perform any analytical fitting procedure to derive a specific curve, to investigate the analytical dependencies of LET$_d$ on $y_{1D}$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{Fig5_RBE_PDD_AAPM2019.pdf}\\\\\n\\noindent\n\\caption{\nShown RBE calculated using variable $\\langle\\langle l \\rangle\\rangle$ (red dots) and constant chord length equal to $5\\mu m$ (black dots), as recommended by ICRU 36.\nRBE calculated after a surface fitting in 3D space to dose, LET and SF data for NSCLC H1437 cell lines.\nA significant non-linear increase in RBE by a factor of $\\approx 4$ in domains distal to the Bragg peak can be captured if $\\langle\\langle l \\rangle\\rangle$ is considered a variable of depth.\nThe biological endpoint used for calculation of RBE, corresponds to 10\\% cell-survival fraction, as reported by Guan {\\em et al.}~[\\onlinecite{Guan2015:SR}].\n}\n\\label{fig4}\n\\end{center}\\vspace{-0.5cm}\n\\end{figure}\n\nWe now turn to present RBE of in-{\\it vitro} clonogenic cell survival assay of H460 cell lines, a type of non-small cell lung cancer (NSCLC).\nThe dependence of SF on dose and LET were measured experimentally in our group~[\\onlinecite{Guan2015:SR}]\nand fitted by 3D global fitting method by the present authors~[\\onlinecite{Abolfath2013:PMB,Abolfath2016:MP,Abolfath2017:SR,Abolfath2019:EPJD}].\nFig.~\\ref{fig4} shows the result of RBE calculated using variable $\\langle\\langle l \\rangle\\rangle$ (red dots) and constant $\\langle\\langle l \\rangle\\rangle = 5\\mu m$ (black dots),\ncorresponding a biological endpoint with 10\\% cell-survival fraction, as reported by Guan {\\em et al.}~[\\onlinecite{Guan2015:SR}].\nThe numerical values of the coefficients in the polynomial expansion of $\\alpha$ and $\\beta$ on $\\Delta_l$ and LET are given by Eqs.(\\ref{eq5_1}-\\ref{eq5_2}).\nFor $LET \\leq 10 keV\/\\mu m$ we obtained the optimal fitting to the following polynomials with\n$N=2$, $b_{1,1} = 0.05237$, $b_{2,1} = 0.0151$ and $b_{2,2} = 0.03265$, corresponding to reduced $\\chi^2 = 0.00541$, $R^2$(COD)$=0.98233$ and adjusted $R^2 = 0.98233$.\nThis is a limit that dependence of $\\alpha$ on $\\Delta_l$ and LET is linear whereas $\\beta$ shows no dependence on $\\Delta_l$ (and LET),\nNote that this is a limit frequently used in literature for all range of LET\n(see e.g., Eq.(II.28) in Ref.~[\\onlinecite{Hawkins1998:MP}] or Eq. (8) in Ref.~[\\onlinecite{Kase2006}]).\nFor $LET > 10 keV\/\\mu m$, we obtained $N=6$ with\n$b_{1,1} = 0.12288$, $b_{2,1} = 0.00624$,\n$b_{3,1} = 1.322\\times 10^{-6}$, $b_{4,1} = 9.449\\times 10^{-7}$,\n$b_{5,1} = 8.812\\times 10^{-7}$,\n$b_{6,1} = 7.725\\times 10^{-8}$, and\n$b_{2,2} = 0.00382$, $b_{3,2} = 0.0007821$, $b_{4,2} = 9.275\\times 10^{-5}$, $b_{5,2} = 9.511\\times 10^{-7}$,\n$b_{6,2} = 1.68\\times 10^{-7}$\ncorresponding to\nreduced $\\chi^2 = 0.05472$, $R^2$(COD)$=0.96777$ and adjusted $R^2 = 0.93911$.\n\n\n\nThe latter is a method of calculation, recommended by ICRU 36.\nA comparison between two methods (a) considering variability in $\\langle\\langle l \\rangle\\rangle$ as a function of depth by double averaging over collisions and tracks and (b) considering geometrical chord length equal to the cell line thickness, $5\\mu m$, evidently exhibits significant difference in predicting RBE by a factor of 4 in domains distal and proximal to the Bragg peak.\n\nSimilar variabilities in RBE as a function of proton range were reported in other experimental works using conventional microdosimetry approach and a methodology based on an empirical or phenomenological ``biological waiting functions\" in calculation of RBE [\\onlinecite{DeNardo2004:RPD}].\nAccordingly, the biological waiting functions fitted to the spectrum of a spread-out Bragg peaks (SOBP) beam of proton collected by a 2.3 mm a microdosimetric prob, using a tissue-equivalent proportional counter (TEPC), resulted in a monotonic increase in RBE up to 2.5, a value close to the RBE reported in this work.\nWe note that the proposed modifications presented in this work, is free from such phenomenological convolution between the beam spectrum and RBE.\n\n\\section{Discussion and conclusion}\nThere are substantial evidence that proton (similar to heavier charged particles) RBE increases as a function of depth.\nThis is partly due to increase in spatial density of collisions as protons pass through tissue and lose energy at an increasing rate.\nTo make connection between experimental radiobiological data and mechanistic models, it is customary approach in literature to use microdosimetry models, developed by Rossi, Kellerer and colleagues~[\\onlinecite{Kellerer1971:RR,Kellerer_Rossi1972:CTRR,Kellerer1984:RR,Kellerer1985:Book,Kellerer1975:REB}].\nEmbedded in these models, geometrical chord length of a SVOI plays a crucial role.\nWe discussed that although this quantity is a good parameter to describe microdosimetry processes and charge collections in tissue proportional chambers, but because it is intrinsically character of the surface and geometrical boundaries of cell nuclei, it cannot directly describe nano-meter scale localities in stochastic microscopic physio-chemical processes in DNA materials and DNA-damage.\nThus we proposed the diffusive mean-free path length of the particle tracks as a new metric that describes appropriately variabilities in collisions compactness as a function of depth in tissue to substitute the geometrical chord-lengths considered in micro-dosimetry modelings.\nThis variation of micro-dosimetry is potentially more appropriate for radio-biological studies.\n\n{\\bf Acknowledgement:}\nThe authors would like to acknowledge useful discussion and scientific exchanges with Drs. Alejandro Carabe-Fernandez and Alejandro Bertolet Reina.\nThe work at the University of Texas, MD Anderson Cancer Center was supported by the NIH \/ NCI under Grant No. U19 CA021239.\n\n\\noindent{\\bf Authors contributions:}\nRA: wrote the main manuscript, prepared figures, performed mathematical derivations and computational steps including Geant4 and Geant4-DNA Monte Carlo simulations and three dimensional surface fitting to the experimental data.\nYH: contributed to Geant4 Monte Carlo simulations and writing the manuscript.\nDC, RS, DG and RM: wrote the main manuscript, contributed to scientific problem and co-supervised the project.\n\n\\noindent{\\bf Corresponding Authors:}\\\\\n$^\\dagger$ ramin1.abolfath@gmail.com \/ Ramin.Abolfath@pennmedicine.upenn.edu \\\\\n$^*$ rmohan@mdanderson.org\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn recent years experiments with strongly-interacting cold atomic gases\nhave attracted much attention.\\cite{ReviewBloch2008}  A particular\nadvantage of these systems is that their parameters can be controlled to a\nhigh degree either directly or via oscillating forces that lead to\nsynthetic gauge fields.\\cite{Creffield2011a, Dalibard2011a}  This allows a flexible\nengineering and simulation of many-body Hamiltonians.  For a theoretical\ndescription, one frequently employs the Hubbard model.  Despite its seeming\nsimplicity, it captures a great variety of condensed-matter phenomena\nranging from metallic behavior to insulators, magnetism, and\nsuperconductivity.\n\nIn the strongly interacting limit of the Hubbard model, particles occupying\nthe same lattice site can bind together, even for repulsive interactions.\nThis occurs when the onsite interaction is much larger than the tunneling \nsuch that energy conservation inhibits the decay into a\nstate with two distant particles.  In principle, both bosons\n\\cite{Valiente2008, Compagno2017} and fermions \\cite{BookEssler2005} can\nform such $N$-particle states.  While the former allow any occupation\nnumber, for fermions with spin $s$, the occupation of one site is\nrestricted to at most $2s+1$ particles. \nIn particular, two spin-1\/2 fermions may reside in a singlet spin \nconfiguration on one lattice site and, thus, form a doublon. Over the last \nyears, they have been investigated both theoretically \n\\cite{Hofmann2012, Bello2016, Bello2017a} and experimentally \n\\cite{Winkler2006, Folling2007, Strohmaier2010, Preiss2015} \nwith cold atoms in optical lattices.\n\nIn the context of solid-state based quantum information and quantum\ntechnologies, arrays of tunnel coupled quantum dots represent a recent platform\nfor similar experiments with electrons.\\cite{Puddy2015a, Zajak2016a}  In\ncomparison to optical lattices, however, these systems are way more\nsensitive to decoherence and dissipation stemming from the interaction with\nenvironmental degrees of freedom such as phonons or charge and current\nnoise.  Since environments may absorb energy, the separation of two\nelectrons in a doublon state is no longer energetically forbidden.  In this\npaper we cast some light on this issue by studying the life times of\ndoublons in a one-dimensional lattice in the presence of charge and current\nnoise, as is sketched in Fig.~\\ref{fig:setup}.  For the environment we\nemploy a Caldeira-Leggett model, \\cite{Leggett1987a, Hanggi1990a} where\ndepending on the type of noise, the bath couples locally to the onsite\nenergies or to the tunnel matrix elements.\n\n\\begin{figure}[t]\n    \\centering\\includegraphics{figure1.pdf}\n    \\caption{Tight-binding lattice occupied by two electrons.  The\n    initial state with a doubly occupied site (doublon) may decay dissipatively\n    into a single-occupancy state with lower energy.  The released energy\n    is of the order of the onsite interaction $U$ and will be absorbed by\n    heat baths representing environmental charge and current noise.\n\\label{fig:setup}\n}\n\\end{figure}\n\nIn Sec.~\\ref{sec:model}, we specify our model and sketch the derivation of\na Bloch-Redfield master equation for the dissipative dynamics.  Section\n\\ref{sec:charge} is devoted to the influence of charge noise, while the\nresults for current noise are worked out in Sec.~\\ref{sec:current}.\nBoundary effects and experimental consequences are discussed in\nSec.~\\ref{sec:discussion}, while the appendix contains details of the\nmaster equation and the averaging of decay rates.\n\n\\section{Model and master equation}\n\\label{sec:model}\n\nThe Fermi-Hubbard model considers particles on a lattice with nearest\nneighbor tunneling and onsite interaction.  For electrons, its\nHamiltonian reads\n\\begin{align}\n    H_S & = -J\\sum\\limits_{j=1}^{N-1}\\sum\\limits_{\\sigma=\\uparrow, \\downarrow}\n    \\left( c^\\dagger_{j+1\\sigma}c_{j\\sigma} + \\mathrm{H.c.}\\right)  \n+ U\\sum\\limits_{j=1}^{N} n_{j\\uparrow}n_{j\\downarrow} \\nonumber \\\\\n& \\equiv -J T + UD \\,,\n\\label{eq:hubbard}\n\\end{align}\nwith the hopping matrix element $J$ and the interaction strength $U$.  The\nfermionic operator $c^\\dagger_{j\\sigma}$ creates an electron with spin\n$\\sigma$ on site $j$, while $n_{j\\sigma}$ is the corresponding number\noperator.  For convenience, we define the hopping operator between \nsites $j$ and $j+1$, as $T_j = \\sum_\\sigma c^\\dagger_{j+1\\sigma}c_{j\\sigma} +\n\\mathrm{H.c.}$\nWhile the Hamiltonian~\\eqref{eq:hubbard} has open boundary conditions,\nwe will also study the case of periodic boundary conditions (ring\nconfiguration) by adding the corresponding term for the hopping between\nthe first and the last site.\n\nHenceforth, we focus on the case of two fermions forming a spin singlet.\nThen we work in a Hilbert space that contains two types of states,\n\\textit{single-occupancy states}\n\\begin{equation}\n\\label{single}\n \\frac{1}{\\sqrt{2}} (c^\\dagger_{i\\uparrow}c^\\dagger_{j\\downarrow}-\n c^\\dagger_{i\\downarrow}c^\\dagger_{j\\uparrow})\\ket{0} \\ , \n \\quad 1\\leq i < j \\leq N \\ , \n\\end{equation}\nand the \\textit{double-occupancy states}, known as \\textit{doublons},\n\\begin{equation}\n\\label{double}\n c^\\dagger_{j\\uparrow}c^\\dagger_{j\\downarrow}\\ket{0} \\ , \\quad j=1,\\cdots,N \\,. \n\\end{equation}\nBoth kinds of states are eigenstates of the operator $D$, which in the\nHilbert space considered is equal to the projector onto the doublon\nstates~\\eqref{double}, in the following denoted as $P_D$.\n\nWhile being different from the states in Eqs.~\\eqref{single} and\n\\eqref{double}, for sufficiently large values of $U$, the eigenstates of \n$H_S$ also discern into two groups, namely $N(N-1)\/2$ states with energies \n$|\\epsilon_n|\\lesssim 4J$ and $N$ states, with energies $|\\epsilon_n|\\approx U$.\nWe will refer to the two groups as the \\textit{low-energy subspace} \n$\\mathcal{H}_0$, and the span of the latter as the \\textit{high-energy \nsubspace} $\\mathcal{H}_1$. In the strongly-interacting regime with $U\\gg J$,\ntreating the tunneling term as a perturbation, it is possible to express the \nprojector onto the high-energy subspace $P_1$ as a power series in $J\/U$,\nsee Ref.~\\onlinecite{MacDonald1988},\n\\begin{equation}\n    P_1 = P_D - \\frac{J}{U}(T^+ + T^-) + \n    \\mathcal{O}\\left( \\frac{J^2}{U^2} \\right) \\ , \\label{eq:projector}\n\\end{equation}\nwhere ${T^+=P_DT(\\mathbb{I}-P_D)}$ and ${T^-=(\\mathbb{I}-P_D)TP_D}$\ncomprise the hopping processes that increase and decrease the double\noccupancy respectively. $\\mathbb{I}$ is the identity operator.\n\nA key ingredient to our model is the coupling to environmental degrees of\nfreedom described as $N$ independent baths of harmonic oscillators,\n\\cite{Leggett1987a,Hanggi1990a}\n${H_B=\\sum_{j,n}\\omega_n a^\\dagger_{jn} a_{jn}}$.  They couple to the\nFermi-Hubbard chain via the Hamiltonian ${H_{SB}=\\sum_j X_j \\xi_j}$,\nwhere the $X_j$ are system operators that will be specified below.\nFor ease of notation, we introduce the collective bath coordinates\n${\\xi_j=\\sum_n g_n (a^\\dagger_{jn}+a_{jn})}$.  Moreover, we assume that all\nbaths are equal and statistically independent, such that\n${\\mean{\\xi_i(t),\\xi_j(t')}= 2S(t-t')\\delta_{ij}}$.\n\nAssuming weak coupling and Markovianity, the time evolution of the \nsystem's density matrix $\\rho$, can be suitably described by a master equation \nof the form \\cite{Redfield1957a,Breuer2007}\n\\begin{align}\n\\label{BlochRedfield}\n    \\dot{\\rho} & = -i[H_S,\\rho]-\\sum_j [X_j,[Q_j,\\rho]]- \n    \\sum_j [X_j,\\{R_j,\\rho\\}] \\\\\n    & \\equiv -i[H_S,\\rho] + \\mathcal{L}[\\rho] \\ .\n    \\nonumber\n\\end{align}\nwith the operators\n\\begin{align}\n  Q_j & = \\frac{1}{\\pi}\\int_0^\\infty d\\tau\\int_0^\\infty d\\omega \n  \\mathcal{S}(\\omega)\\tilde{X_j}(-\\tau) \\cos \\omega\\tau \\ , \\label{eq:R}\\\\\n  R_j &= \\frac{-i}{\\pi}\\int_0^\\infty d\\tau\\int_0^\\infty d\\omega \n  \\mathcal{J}(\\omega)\\tilde{X_j}(-\\tau) \\sin \\omega\\tau \\ . \\label{eq:Q}\n\\end{align}\nThe tilde denotes the interaction picture with respect to the system\nHamiltonian, ${\\tilde{X}_j(-\\tau)=e^{-i H_S\\tau}X_je^{i H_S\\tau}}$, while\n${\\mathcal{J}(\\omega)=\\pi\\sum_n |g_n|^2 \\delta(\\omega- \\omega_n)}$ is the\nspectral density of the baths and\n${\\mathcal{S}(\\omega)=\\mathcal{J}(\\omega)\\coth(\\beta\\omega\/2)}$ is the\nFourier transformed of the symmetrically-ordered equilibrium\nautocorrelation function $\\mean{\\{\\xi_j(\\tau),\\xi_j(0)\\}}\/2$.\n$\\mathcal{J}(\\omega)$ and $\\mathcal{S}(\\omega)$ are independent of $j$\nsince all baths are identical. We will assume an ohmic spectral density\n${\\mathcal{J}(\\omega)=\\pi\\alpha\\omega\/2}$, where the dimensionless parameter\n$\\alpha$ characterizes the dissipation strength.\n \n\n\\section{Charge noise}\n\\label{sec:charge}\n\nFluctuations of the background charges in the substrate essentially act\nupon the charge distribution of the chain. Therefore, we model it by\ncoupling the occupation of each site to a heat bath, such that\n\\begin{equation}\nH_{SB}^Q = \\sum_{j,\\sigma} n_{j,\\sigma} \\xi_j \\,,\n\\end{equation}\nwhich means $X_j = n_j$. This fully specifies the master equation~\\eqref{BlochRedfield}.\n\nTo get a qualitative impression of the decay dynamics of a doublon, let us\nstart by discussing the time evolution of a doublon state in the strongly\ninteracting regime shown in Fig.~\\ref{fig:dynamics}.  For $\\alpha=0$,\ni.e., in the absence of dissipation, the two electrons will essentially\nremain together throughout time evolution. This is due to energy\nconservation and the fact that kinetic energy in a lattice is bounded, it\ncan be at most $2|J|$ per particle. Thus, particles forming a doublon\ncannot split, as they would not have enough kinetic energy on their own to\ncompensate for the large $U$.  However, since the doublon states are not\neigenstates of the system Hamiltonian, we observe some slight oscillations\nof the double occupancy $\\langle D\\rangle$.  Still the time average of\nthis quantity stays close to unity, see Fig.~\\ref{fig:dynamics}(a). \n\nOn the contrary, if the system is coupled to a bath, doublons will be able to \nsplit releasing energy into the environment.  Then the density operator\neventually becomes the thermal state $\\rho_\\infty\\propto e^{-\\beta H_S}$.\nDepending on the temperature and the interaction strength, the\ncorresponding asymptotic doublon occupancy $\\mean{D}_\\infty$ may still\nassume an appreciable value.\n\n\\begin{figure}\n    \\centering\\includegraphics{figure2.pdf}\n    \\caption{Time evolution of the double occupancy in a system with charge\n        noise. The initial state consists of a doublon localized in a \n        particular site of a chain with periodic boundary conditions.  \n        Parameters: $N=5$, $U=10J$ and $\\alpha=0.04$. (a) Comparison between \n        free dynamics ($\\alpha=0$) and dissipative dynamics ($\\alpha\\neq 0$). \n        Temperature is set to $k_B T=0.01J$. The green line corresponds to the \n        occupancy of the high-energy subspace for the case with $\\alpha\\neq 0$ \n        and illustrates the bound given in \\eqref{eq:bound}. (b) Decay of the \n        high-energy subspace occupancy for different temperatures ranging from  \n        $0.01J$ to $1000J$. The slope of the curves at time $t=0$ is the same \n        in all cases and coincides with the value given by \\eqref{eq:avergamma} \n        (red dashed line).} \n        \\label{fig:dynamics} \n\\end{figure} \n\n\\subsection{Numerical analysis \\label{sec:numerics}}\n\nTo gain quantitative insight, we decompose our master equation\n\\eqref{BlochRedfield} into the system eigenbasis and obtain a form\nconvenient for numerical treatment (for details, see\nAppendix~\\ref{app:masterequation}). A typical time evolution of the occupancy\n$\\langle D\\rangle$ is shown in Fig.~\\ref{fig:dynamics}(a).  It exhibits an\nalmost mono-exponential decay, such that the doublon life time $T_1$ can be\ndefined as the $1\/e$ time of the difference between initial and final\nvalue of $\\langle D\\rangle$,\n\\begin{equation}\n    \\frac{\\mean{D}_{T_1}-\\mean{D}_\\infty}{1-\\mean{D}_\\infty}=\\frac{1}{e} \\,.\n    \\label{eq:numericgamma}\n\\end{equation}\n\nThe corresponding decay rate $\\Gamma = 1\/T_1$ is shown in\nFig.~\\ref{fig:numerics_occup} as a function of the temperature for different\nvalues of the dissipation strength $\\alpha$. For small $\\alpha$ and\nintermediate temperatures, $\\Gamma$ increases with the temperature,\nreaching a maximum after which the tendency inverts.  For sufficiently\nlarge temperatures, $\\Gamma \\propto (\\alpha k_B T)^{-1}$.\n\n\\begin{figure}[tb] \n    \\centering\\includegraphics{figure3.pdf} \n    \\caption{Temperature dependence of the numerically obtained decay rate \n        for a chain with $N=5$ and periodic boundary conditions in the presence\n        of charge noise. The interaction energy is set to $U=10J$. For values \n        of the coupling strength $\\alpha\\lesssim 0.04$ we obtain approximately \n        the same curve (continuous line). Inset: Values for $\\Gamma$ on a \n        logarithmic scale demonstrating the proportionality \n        $\\propto 1\/\\alpha T$.} \n\\label{fig:numerics_occup}\n\\end{figure}\n\n\\subsection{Analytical estimate for the decay rate}\n\nAn analytical estimate for the decay rates can often be gained from the\nbehavior at the initial time $t=0$, i.e.\\ from $\\dot\\rho(0) =\n-i[H_S,\\rho_0]+ \\mathcal{L}\\rho_0$ with $\\rho_0=\\rho(0)$ being the pure\ninitial state.  In the present case, however, the calculation is hindered\nby the fast initial oscillations witnessed in Fig.~\\ref{fig:dynamics}(a).\nThese oscillations stem from the mixing of the doublon states with the\nsingle-occupancy states.  To circumvent this problem, we focus for the\npresent purpose on the occupancy of the high-energy subspace, $\\langle\nP_1\\rangle$ shown in Fig.~\\ref{fig:dynamics}(b).  It turns out that this\nquantity evolves more smoothly while it decays also on the time scale\n$T_1$.  The reason for its lack of fast oscillations is that the projector\n$P_1$ commutes with the system Hamiltonian, so that it expectation value is\ndetermined solely by dissipation.  Notice that the initial decay is\ntemperature independent, while at a later stage, the decay is strongest for\nintermediate temperatures.\n\nA formal way of understanding the similarity of the long time dynamics of\n$\\langle D\\rangle$ and $\\langle P_1\\rangle$ is provided by the estimate\n\\begin{align}\n    |\\tr{P_1\\rho}-\\tr{D\\rho}|\n    \\leq {}& \\sqrt{2}\\|\\rho\\|\\sqrt{N-\\tr{P_1P_D}} \\\\\n    \\simeq {}& 2\\sqrt{2N}J\/U \\ ,\n    \\label{eq:bound}\n\\end{align}\nwhere the first lines follows from the Cauchy-Schwarz inequality for the\ninner product of operators, $(A,B)=\\tr{A^\\dagger B}$,\nwhile the second line stems from the\nperturbative expansion of $P_1$ given by Eq.~\\eqref{eq:projector}.  The\nresult implies that when neglecting corrections of the order of $J\/U$, we may\ndetermine $T_1$ and $\\Gamma$ from either quantity.  Nevertheless it is\ninstructive to analytically evaluate $\\Gamma$ for the decay of both\n$\\langle D\\rangle$ and $\\langle P_1\\rangle$.\n\nFollowing our hypothesis of a mono-exponential decay, we expect\n\\begin{equation}\n    \\label{eq:P1decay}\n    \\mean{P_1}\\simeq \\Delta e^{-\\Gamma t} + \\mean{P_1}_\\infty \\ ,\n\\end{equation} \ntherefore,\n\\begin{equation}\n    \\Gamma\\simeq - \\frac{1}{\\Delta}\n    \\left. \\frac{d \\mean{P_1}}{d t}\\right|_{t=0}=\n    -\\frac{\\tr{P_1\\mathcal{L}[\\rho_0]}}{\\mean{P_1}_0-\\mean{P_1}_\\infty} \\ .\n    \\label{eq:gamma}\n\\end{equation}\nThis expression still depends slightly on the specific choice of the\ninitial doublon state, in particular for open boundary conditions (see\nSec.~\\ref{sec:boundary}, below).  To obtain a more global picture, we\nconsider an average over all doublon states, which can be performed\nanalytically.\\cite{Storcz2005a}  From Eq.~\\eqref{eq:gamma}, we find the\naverage decay rate\n\\begin{multline}\n    \\overline{\\Gamma}=\\frac{1}{N\\Delta}\\sum_j \\tr{P_D[Q_j,[X_j,P_1]]} \\\\\n    -\\tr{P_D\\{R_j,[X_j,P_1]\\}} \\ .\n    \\label{eq:avergamma}\n\\end{multline}\nFor details of the averaging procedure, see Appendix~\\ref{app:average}.\n\nFor a further simplification, we have to evaluate the expressions\n\\eqref{eq:R} and \\eqref{eq:Q} which is possible by approximating the\ninteraction picture coupling operator as $\\tilde{X}_j(-\\tau)\\simeq X_j\n-i\\tau[H_S,X_j]$.  This is justified as long as the decay of the\nenvironmental excitations is much faster than the typical system evolution,\ni.e., in the high-temperature regime (HT).  Inserting our approximation for\n$\\tilde{X}_j$ and neglecting the imaginary part of the integrals, we arrive at\n\\begin{align} \n    Q_j & \\simeq \\frac{1}{2}\\lim_{\\omega\\rightarrow 0^+} \n    \\mathcal{S}(\\omega) X_j = \\frac{\\pi}{2}\\alpha k_B T X_j\\ , \\\\ R_j & \\simeq -\n    \\frac{1}{2}\\lim_{\\omega\\rightarrow 0^+} \\mathcal{J}'(\\omega) [H_S,X_j] = \n    \\frac{\\pi}{4}\\alpha [H_S,X_j] \\ .  \n\\end{align} \nWith these expressions, Eq.~\\eqref{eq:avergamma} results in a temperature\nindependent decay rate.  Notice that any temperature dependence stems from\nthe $Q_j$ in the first term of Eq.~\\eqref{eq:avergamma} which vanishes in\nthe present case.  While this observation agrees with the numerical findings in\nFig.~\\ref{fig:dynamics} for very short times, it does not reflect the\ntemperature dependent decay of $\\langle P_1\\rangle$ at the more relevant\nintermediate stage.\n\nThis particular behavior hints at the mechanism of the bath-induced doublon\ndecay.  Let us notice that the coupling to charge noise, $X_j=n_j$,\ncommutes with $D$.  Therefore, the initial state is robust against the\ninfluence of the bath.  Only after mixing with the single-occupancy\nstates due to the coherent dynamics, the system is no longer in an\neigenstate of the $n_j$, such that decoherence and dissipation become\nactive.  Thus, it is the combined action of the system's unitary evolution\nand the effect of the environment which leads to the doublon decay.\n\nAn improved estimate of the decay rate, can be calculated by averaging the\ntransition rate of states from the high-energy subspace to the low-energy\nsubspace.  Let us first focus on regime $k_BT\\gtrsim U$ in which we can\nevaluate the operators $Q_j$ in the high-temperature limit.  Then the\naverage rate can be computed using expression \\eqref{eq:avergamma} and\nreplacing $P_D$ by $P_1$, see Appendix \\ref{app:masterequation}. With the\nperturbative expansion of $P_1$ in Eq.~\\eqref{eq:projector} we obtain to\nleading order in $J\/U$ the averaged rate\n\\begin{equation} \n    \\overline{\\Gamma}_\\mathrm{HT} \\simeq \n    \\frac{4\\pi\\alpha J^2}{U^2 \\Delta}\\left(2k_B T + U \\right) \\,,\n    \\label{eq:GOHT}\n\\end{equation}\nvalid for periodic boundary conditions.  For open boundary conditions, the\nrate acquires an additional factor $(N-1)\/N$.  Notice that we have\nneglected back transitions via thermal excitations from singly occupied\nstates to doublon states.  We will see that this leads to some smaller\ndeviations when the temperature becomes extremely large.  Nevertheless, we\nrefer to this case as the high-temperature limit.\n\nIn the opposite limit, for temperatures ${k_B T < U}$, the decay rate\nsaturates at a constant value. To evaluate $\\overline{\\Gamma}$ in this limit, it\nwould be necessary to find an expression for $\\tilde{X}_j(-\\tau)$ dealing\nproperly with the $\\tau$-dependence for evaluating the noise kernel, a\nformidable task that may lead to rather involved expressions.\nNevertheless, one can make some progress by considering the transition of\none initial doublon to one particular single-occupancy state.  This\ncorresponds to approximating our two-particle lattice model by the\ndissipative two-level system for which the decay rates in the Ohmic case\ncan be taken from the literature,\\cite{Weiss1989a, Makhlin2001b} see\nAppendix \\ref{app:TLS}.  Relating $J$ to the tunnel matrix element of the\ntwo-level system and $U$ to the detuning, we obtain from\nEq.~\\eqref{app:Gammaii} the temperature-independent expression\n\\begin{equation}\n    \\overline{\\Gamma}_\\text{LT}  \\simeq \\frac{8\\pi\\alpha J^2}{U \\Delta} \\ ,\n\\label{eq:GOLT}\n\\end{equation}\nwhich formally corresponds to Eq.~\\eqref{eq:GOHT} with the temperature set\nto $k_B T=U\/2$.\n\nFigure~\\ref{fig:analyticsOccup} provides a comparison of these analytical\nfindings with numerical results.  The data in panel (a) reveal that the\ntransition between the low-temperature regime and the high-temperature\nregime is rather sharp and occurs at $U\\approx k_BT$.  Panel (b) shows\n$\\Gamma$ as a function of the temperature.  For low temperatures, the\nnumerical values saturate at $\\overline{\\Gamma}_\\text{LT}$ obtained from the\napproximate mapping to a two-level system.  For high temperatures, the\nanalytical prediction $\\overline{\\Gamma}_\\text{HT}$ seems slightly too \nlarge.  The discrepancy stems from neglecting thermal excitations, as \nmentioned above.\n\n\\begin{figure}[tb]\n    \\centering\\includegraphics{figure4.pdf}\n    \\caption{Comparison between the numerically computed decay rate and the \n        analytic formulas \\eqref{eq:GOHT} and \\eqref{eq:GOLT} for a chain\n        with $N=5$ sites and periodic boundary conditions in the case of \n        charge noise.  The dissipation strength is $\\alpha=0.02$.\n        (a) Dependence on the interaction strength for a fixed temperature $k_B \n        T=20J$. (b) Dependence on the temperature for a fixed interaction \n        strength $U=20J$.}\n    \\label{fig:analyticsOccup}\n\\end{figure}\n\n\\section{Current noise}\n\\label{sec:current}\n\nFluctuating background currents mainly couple to the tunnel matrix elements\nof the system.  Then the system-bath interaction is given by setting \n$X_j=T_j$ and reads\n\\begin{equation}\nH_{SB}^I = \\sum_{j,\\sigma} (c^\\dagger_{j+1\\sigma}c_{j\\sigma} \n+ c^\\dagger_{j\\sigma} c_{j+1\\sigma} ) \\xi_j \\,.\n\\end{equation}\nDepending on the boundary conditions, the sum may include the term with\n$j=N$.  The main qualitative difference of this choice is that in contrast\nto charge noise, $H_{SB}^I$ does not commute with the projector to the\ndoublon subspace and, thus, generally ${\\tr{D\\mathcal{L}[\\rho]}\\neq 0}$.\nThis enables a direct dissipative decay without the detour via an admixture\nof single-occupancy states to the doublon states.  As a consequence,\nfor the same value of the dimensionless dissipation parameter $\\alpha$, the\ndecay may be much faster.  Also the temperature dependence of the decay\nchanges significantly, as can bee seen in Fig.~\\ref{fig:numerics_hopp}.\nWhile $\\overline{\\Gamma}$ is still proportional to $\\alpha$, it now grows\nmonotonically with the temperature.\n\n\\begin{figure}\n    \\centering\\includegraphics{figure5.pdf}\n    \\caption{Average decay rate of the doublon states under the influence\n    of current noise for various dissipation strengths as a function of\n    the temperature.  The chain consists of $N=5$ sites with periodic boundary\n    conditions, while the interaction is $U=10J$.}\n    \\label{fig:numerics_hopp}\n\\end{figure}\n\nAs in the last section, we proceed by calculating analytical estimates for\nthe decay rates.  However, since the time evolution is no longer\nmono-exponential (not shown), we no longer start from the\nansatz~\\eqref{eq:avergamma}, but estimate the rate from the slope of the\noccupancy $\\langle P_1\\rangle$ at initial time,\n\\begin{equation}\n    \\Gamma \\simeq - \\left. \\frac{d \\mean{P_1}}{d t} \\right|_{t=0} =\n        - \\tr{P_1\\mathcal{L}[\\rho_0]} \\ .\n    \\label{eq:decay_hopp}\n\\end{equation}\nWe again perform the average over all doublon states for $\\rho_0$\nin the limits of high and low temperatures.  For periodic boundary\nconditions, we obtain to lowest order in $J\/U$ the high and low temperature\nrates \n\\begin{align}\n    \\overline{\\Gamma}_\\mathrm{HT} & = 2\\pi\\alpha \\left(2k_B T+U \\right) \\,,\n    \\label{eq:GTHT}  \\\\\n    \\overline{\\Gamma}_\\text{LT} & = 4\\pi\\alpha U \\,,\n    \\label{eq:GTLT}\n\\end{align}\nrespectively,\nwhile open boundary conditions lead to the same expressions but with a\ncorrection factor $(N-1)\/N$.\nIn Fig.~\\ref{fig:analyricsHopp}, we compare these results with the\nnumerically evaluated ones as a function of the interaction\n[Fig.~\\ref{fig:analyricsHopp}(a)] and\nthe temperature [Fig.~\\ref{fig:analyricsHopp}(b)].  Both show that the\nanalytical approach correctly predicts the (almost) linear behavior at\nlarge values of $U$ and $k_BT$, as well as the saturation for small values.\nHowever, the approximation slightly overestimates the influence of the\nbath.\n\n\\begin{figure}\n    \\centering\\includegraphics{figure6.pdf}\n    \\caption{Numerically obtained decay rate in comparison with the\n        approximations~\\eqref{eq:decay_hopp}, \\eqref{eq:GTHT} and \n        \\eqref{eq:GTLT} for a chain with $N=5$ sites and periodic boundary \n        conditions in the case of current noise with strength $\\alpha=0.02$.  \n        The results are plotted as a function of (a) the interaction and the \n        temperature $k_B T=20J$ and (b) for a fixed interaction $U=20J$ as a \n        function of the temperature.}\n    \\label{fig:analyricsHopp}\n\\end{figure}\n\nWhile the rates reflect the decay at short times, it is worthwhile to\ncomment on the long time behavior under the influence of current\nnoise.  For open chains as well as for closed chains with an even\nnumber of sites, it is not ergodic as the long-time solution is not unique.\nThe reason for this is the existence of a doublon state\n${\\ket{\\Phi}=\\frac{1}{\\sqrt{N}}\\sum_{j=1}^N (-1)^j\nc^\\dagger_{j\\uparrow}c^\\dagger_{j\\downarrow}\\ket{0}}$ which is an\neigenstate of $H_S$ without any admixture of single-occupancy states.\nSince $T_j\\ket{\\Phi}=0$ for all sites $j$, current noise may affect\nthe phase of ${\\ket{\\Phi}}$, but cannot induce its dissipative decay.\nFor a closed chain with an odd number of sites, by contrast, the alternating\nphase of the coefficients of $\\ket{\\Phi}$ is incompatible with periodic\nboundary conditions, unless a flux threatens the ring.  As a consequence,\nthe chain eventually resides in the thermal state $\\propto\\exp(-\\beta\nH_S)$.  The difference is manifest in the final value of the doublon\noccupancy at low temperatures.  For closed chains with an odd number of\nsites, it will fully decay, while in the other cases, the population of\n$\\ket{\\Phi}$ will survive.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\\subsection{Dimension and boundary effects}\n\\label{sec:boundary}\n\n\\begin{figure}[tb]\n    \\centering\\includegraphics{figure7.pdf}\n    \\caption{Decay rates of the double occupancy for a chain with $N=5$ sites \n        with open boundary as a function of the initial location of the doublon.\n\tThe values for $\\Gamma$ are taken as the inverse of the $T_1$ time\n\tobtained from a numerical propagation of the master equation.\n        The red dashed line marks the value for closed boundary conditions.\n\tThe other parameters are $U=20J$, $\\alpha=0.01$, $k_B T=5J$.}\n    \\label{fig:site_dependence}\n\\end{figure}\n\nSo far, we have considered decay rates as the averages of all possible\ninitial doublon or high-energy states.  While this is sufficient for a\ngeneric estimate of the life times, it ignores the fact that the behavior\nof individual states may differ significantly, in particular when the\ninitial state is located at a boundary, which reduces the number of\naccessible decay channels.  In Fig.~\\ref{fig:site_dependence} we present\nthe decay rates for doublons as a function of the initial site.  It reveals\nthat in comparison to states at the center, an initial localization at the\nfirst or last site, may double the life time for charge noise and enhance by\nit by a factor three for current noise.  The dashed lines in these plots\nmarks the value for periodic boundary conditions, for which the value is\npractically the same as for a states in the center.\n\nThis knowledge about the role of boundaries and nearest neighbors provides\nsome hint on the doublon life time in higher-dimensional lattices.  Let us\nnotice that Let the decay rates \\eqref{eq:gamma} and \\eqref{eq:decay_hopp} \ncontain one term for each single-occupancy state that is directly tunnel \ncoupled to the initial site.  Assuming that all terms are of the same order, \nwe expect that $\\overline{\\Gamma}$ is by and large proportional to the \ncoordination number of the lattice sites.  Therefore the life time should \ndecrease only moderately with the dimension, roughly as \n$T_1 = \\Gamma^{-1} \\sim 2^{-D}$.  From the data in \nFig.~\\ref{fig:site_dependence}(b), we can appreciate that for current\nnoise, the difference between center and border is even larger.  Thus,\nincreasing dimensionality should have a slightly larger impact on the\ndoublon life times.\n\n\n\\subsection{Experimental implications}\n\\label{sec:experiment}\n\nA current experimental trend is the fabrication of larger arrays of quantum\ndots,\\cite{Puddy2015a, Zajak2016a} which triggered our question on the\nfeasibility of doublon experiments in solid-state systems.  While the size\nof these arrays would be sufficient for this purpose, their dissipative\nparameters are not yet fully known.  For an estimate we therefore consider\nthe values for GaAs\/InGaAs quantum dots which have been determined recently\nvia Landau-Zener interference.\\cite{Forster2014, onalpha}\nNotice, that for the strength of the current noise, only an upper\nbound has been reported.  We nevertheless use this value, but keep in mind\nthat it leads to a conservative estimate.  In contrast to the former\nsections, we now compute the decay for the simultaneous action of charge\nnoise and current noise.\n\n\\begin{figure}\n    \\centering\\includegraphics{figure8.pdf}\n    \\caption{(a) Spatially resolved doublon dynamics in a chain with $N=5$ \n    sites and open boundary conditions for the dissipative parameters\n    determined in Ref.~\\onlinecite{Forster2014}, i.e., for the dissipation\n    strengths\\cite{onalpha} $\\alpha_Q=3\\times 10^{-4}$ and $\\alpha_I=5\\times 10^{-6}$, \n    the tunnel coupling $J=13\\,\\mu\\mathrm{eV}$, interaction $U=1.3$\\,meV, and \n    temperature $T=10$\\,mK.  (b) Corresponding decay of the double occupancy \n    (solid line) and state purity (dashed).}\n    \\label{fig:realistic_dynamics}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\\includegraphics{figure9.pdf}\n    \\caption{Doublon life time as a function of the temperature for\n\tdifferent interaction strengths.  The other parameters are as in\n\tFig.~\\ref{fig:realistic_dynamics}. Vertical dashed lines mark the \n    temperature corresponding to the tunneling energy and the Hubbard \n    interaction energy.  \n    Inset: $T_1$ time for the optimized value of the interaction,\n\t$U =10J =130\\,\\mu$eV and a current noise with\n\t$\\alpha_I = 2\\times10^{-6}$.  The latter is smaller than the value in\n\tFig.~\\ref{fig:realistic_dynamics}, but still realistic.\n\t}\n    \\label{fig:realistic_experiment}\n\\end{figure}\n\nFigure~\\ref{fig:realistic_dynamics}(a) shows the dissipative time evolution\nfor a doublon initially localized at the center of a chain with 5 sites.\nThe dynamics exhibits a few coherent oscillations in which the doublon\nevolves into a superposition of the kind $|2,0,0\\rangle + |0,0,2\\rangle$,\nwhich represents an example of a NOON state.\\cite{Lee2002a}\nEach component propagates to one end of the chain, where is it reflected\nsuch that subsequently the initial states revives.  In\nFig.~\\ref{fig:realistic_dynamics}(b), we depict the evolution of the\ncorresponding doublon occupancy and the purity.  Both quantities decay\nrather smoothly.  This agrees to the finding found in\nSec.~\\ref{sec:current} for pure current noise which obviously dominates.\nIt is also consistent with the values for the respective analytical decay\nrates in the low-temperature limit.  Figure~\\ref{fig:realistic_experiment}\nshows the $T_1$ times for two different interaction strengths.  It reveals\nthat for low temperatures $T\\lesssim J\/k_BT$, the life time is essentially\nconstant, while for larger temperatures, it decreases moderately until\n$k_BT$ comes close to the interaction $U$.  For higher temperatures,\n$\\Gamma$ starts to grow linearly.  On a quantitative level, we expect life\ntimes of the order $T_1\\sim 5$\\,ns already for a moderately low\ntemperatures $T\\lesssim 100$\\,mK.  Since we employed the value of the upper\nbound for the current noise, the life time might be even larger.\n\nConsidering the analytical estimates for the decay rates at low\ntemperatures, Eqs.~\\eqref{eq:GOLT} and \\eqref{eq:GTLT}, separately,\nlets us conclude that for smaller values of $U$, current noise becomes\nless important, while the impact of charge noise grows.  Therefore, a\nstrategy for reaching larger $T_1$ times is to\ndesign quantum dots arrays with smaller onsite interaction, such\nthat the ratio $U\/J$ becomes more favorable.  The largest $T_1$ is expected\nin the case in which both low-temperature decay rates are equal,\n$\\overline{\\Gamma}_\\text{LT,charge} = \\overline{\\Gamma}_\\text{LT,current}$, \nwhich for the present experimental parameters is found at $U\\sim 10J$ (while \nour data is for $U\\sim 100 J$).  This implies that in an optimized device, \nthe doublon life times could be larger by one order of magnitude to reach \nvalues of $T_1\\sim 50$\\,ns, which is corroborated by the data in the inset \nof Fig.~\\ref{fig:realistic_experiment}.\n\n\n\\section{Conclusions}\n\nWe have investigated the life times of double-occupancy states or doublons\nin a one-dimensional Hubbard model under the influence of dissipating\nenvironments.  While in optical lattices, the resulting dissipative decay\nmay be of minor influence, for quantum dot arrays, it will be a limiting\nfactor.\n\nWe have considered two different couplings between the system and its\nenvironment, which physically correspond to the impact of charge noise and\ncurrent noise, respectively.  Within a Bloch-Redfield formalism, this model\ncan be treated with a master equation, which allows one to\nnumerically determine the life times from the time evolution of the reduced\ndensity operator.  Moreover, it provides analytical estimates for the\ninitial decay rates.  It turned out that the striking difference between\nthe two couplings is that the impact of charge noise decreases with the\ninteraction, while current noise becomes increasingly relevant.\n\nFor present quantum dots, the doublon life time is expected to be of the\norder 5\\,ns, which would limit the coherent dynamics to only a few\nperiods.  However, our analytical estimates suggest that for quantum dot\narrays with smaller onsite interaction, an extension by one order of\nmagnitude should be feasible.  Thus, the recent trend towards arrays with\never more coherently coupled quantum dots will allow the experimental\nrealization of effects that so far have been measured only in optical\nlattices.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nShort-pulse, high-intensity laser and beam plasma interaction is an active and robust research area. It involves relativistic, nonlinear and ultrafast plasma physics. It is also a critical topic to the field of plasma based acceleration (PBA). When an intense laser or particle beam propagates through a plasma, it excites a relativistic plasma wave (wakefield). These wakefields support extremely high and coherent accelerating fields which can be more than three orders of magnitude in excess of those in conventional accelerators. The field of PBA has seen rapid experimental progress with many milestones being achieved, including electron acceleration driven by an electron \\cite{blumenfeld2007,litos2014}, laser \\cite{faure2004,geddes2004,mangles2004,gonsalves2011,wang2013,steinke2016,guenot2017} or proton beam \\cite{adli2018}, positron acceleration \\cite{corde2015} and PBA-based radiation generation \\cite{kneip2010,cipiccia2011,nie2018}.\n\nThe rapid progress in experiments has been greatly facilitated by start-to-end simulations using high fidelity particle based methods. The nonlinear aspects of the physics requires the use of fully kinetic tools and the particle-in-cell (PIC) method has proven indispensable. The fully explicit relativistic electromagnetic (EM) PIC method has been used very successfully \\cite{dawson1983,birdsall2005,hockney1988}. In this method, individual macro-particles described by Lagrangian coordinates are tracked in continuous phase space as finite size particles (positions and momentum can have continuous values), and then moments of the distribution (\\emph{e.g.} charge and current density) are deposited onto  stationary mesh\/grid points. The electromagnetic fields are advanced forward in time on the grid points using a discretized version of Maxwell's equations. The new fields are then interpolated to the particles positions to push the particles to new momenta and positions using the relativistic equations of motion. This sequence is repeated for a desired number of time steps.\n\nIn most fully explicit PIC codes, a finite-difference time-domain (FDTD) method is used to advance the time-dependent Maxwell's equations and the differential operators are approximated through a finite difference representation (usually accurate to second order of the cell size).  However, to prevent a numerical instability, the time step is constrained by the Courant-Friedrichs-Lewy (CFL) condition which fundamentally couples the spatial and temporal resolution. Roughly speaking, the time step size needs to be less than the smallest cell size which in turn is determined by the smallest physical scale of interest. A second order representation of the time derivative is then used to push the particles. When modeling short-pulse laser and beam-plasma interactions, the moving window technique \\cite{decker1994} is always used. In this technique, only a finite window that keeps up with the laser is simulated. New cells and fresh plasma are added to the front, while cells and plasma are dropped off the back. This works because no information and physics that has been dropped can effect the plasma in front of it during the simulation.\n\nToday's supercomputers are capable of providing $\\sim10^{16}$ to $\\sim10^{17}$ floating point operations per second \\cite{top500}. To utilize such computers the algorithm needs data structures that permit many thousands of cores to simultaneously push particles. Effective utilization of such computers has enabled full-scale 3D modeling of intense laser or relativistic charged particles interaction with plasma in some cases. However, even with today's computers and PIC software, it is still not possible to carry out start-to-end simulations of every experiment or proposed concept in full 3D using standard PIC codes. In addition, explicit EM PIC codes can be susceptible to numerical issues including the numerical Cerenkov instability (NCI) \\cite{xu2013} and errors to the fields that surround relativistically moving charges \\cite{xu2019}. Furthermore, beam loading studies can require very fine resolution in the transverse direction when ion collapse within a particle beam arises \\cite{rosenzweig2005,gholizadeh2010,an2017}.\n\nVarious methods have been developed to more efficiently model the short-pulse laser and beam-plasma interactions in PBA. These include the boosted frame technique \\cite{vay2007}, the quasi-static approximation \\cite{mora1997,whittum1997,lotov2003,huang2006,an2013,mehrling2014}, and an azimuthal mode expansion method \\cite{lifschitz2009,davidson2015,lehe2016}. The first two are based on the assumption that all relevant waves move forward with velocities near the speed of light, \\emph{e.g.}, no radiation propagates backwards. Some of these methods can be combined \\cite{yu2016}.\n\nThe quasi-static approximation (QSA) was first presented as an analytical tool for studying short-pulse laser interactions \\cite{sprangle1990a,sprangle1990b}. The applicability of QSA originates from the disparity in time\/length scales between how the laser or particle beam evolves and the period\/wavelength of the plasma wake (the plasma response). In the QSA the plasma response is calculated by assuming that the shape of the laser or particle beam (envelope and energy or frequency) is static and the resulting fields from the plasma response are then used to advance the laser or beam forward using a very large time step. It was not until the work of Antonsen and Mora that a PIC algorithm was developed to utilized the QSA. They showed how to push a slice of plasma through a static laser (or move a static laser past a slice of plasma). Their code WAKE \\cite{mora1997} is two dimensional (2D) using r-z coordinates and it can model both lasers and particle beams. Whittam also independently developed a QSA PIC code for modeling particle beam-plasma interaction \\cite{whittum1997}. In this implementation, it was assumed that plasma particles motion is approximated to be non-relativistic so plasma particles do not move in the beam propagation direction. LCODE \\cite{lotov2003} is another 2D r-z PIC code based on the QSA that only models particle beam drivers. QuickPIC \\cite{huang2006,an2013} was the first fully 3D QSA based code and it is fully parallelized including a pipelining parallel algorithm \\cite{feng2009,an2013}. HiPACE \\cite{mehrling2014} is a more recent 3D PIC code based on the QSA. QuickPIC can efficiently simulate both laser pulses and particle beams. It can achieve $10^2$ to $10^4$ speedup without loss of accuracy when compared against fully explicit PIC codes (\\emph{e.g.} OSIRIS \\cite{fonseca2002}) for relevant problems.\n\nAnother recently developed method to enhance the computational efficiency applies the azimuthal Fourier decomposition \\cite{lifschitz2009}. In this method, all the field components and current (and charge) density are expanded into a Fourier series in $\\phi$ in the azimuthal direction (into azimuthal harmonics denoted by $m$); and the series can be truncated at a value of $m$ determined by the degree of asymmetry for the problem of interest. This algorithm can be viewed as a hybrid method where the PIC algorithm is used in r-z grid and a gridless method is used in $\\phi$ and it is sometimes referred to as quasi-3D. By using this algorithm, the problem reduces to solving the complex amplitude (coefficients for Fourier series) for each harmonic on a 2D grid. The complex amplitude, as a function of $r$ and $z$, is updated only at a cost similar to an r-z 2D code. Therefore, if only a few harmonics are kept the algorithm is very efficient. For example, a linearly polarized laser with a symmetric spot size can be described by only the first harmonic. In addition to the much lower cost for advancing fields, much fewer macro-particles are needed for high fidelity. It has been found that speedups of more than two orders of magnitude over a full 3D code are possible.\n\nThe quasi-3D method has been implemented into some fully explicit 3D PIC codes \\cite{lifschitz2009,davidson2015,lehe2016} and used to study laser \\cite{nie2018,ferri2018} and beam \\cite{corde2015} plasma interactions. It also been successfully combined with the boosted frame method \\cite{yu2016}. However, the azimuthal mode expansion has not been combined with the QSA method or implemented into a quasi-static PIC code. If the quasi-3D technique can be successfully combined with the QSA then dramatic speedups will be possible for problems which are nearly azimuthally symmetric. Such a code will greatly extend the scope of PBA research problems that can be studied numerically.\n\nIn this paper, we describe a new code that combines a QSA 3D PIC code with an azimuthal Fourier decomposition, called QPAD (QuickPIC with azimuthal decomposition). The code contains similar procedures and workflow as the 3D quasi-static PIC code QuickPIC, but with the entirely new framework to utilize the azimuthal decomposition. While QuickPIC uses FFT based Poisson solvers to update the fields in each 2D slice of plasma, QPAD computes the fields by means of finite-difference (FD) solvers using the cyclic reduction method \\cite{press2010}.\nWithout loss of accuracy, the code achieves dramatic speedup over fully 3D QuickPIC for a wide range of beam-driven plasma acceleration problems. QPAD currently only supports particle beam drivers.\n\nThe paper is organized as follows. In Section \\ref{sec:theory}, we derive the governing equations for the complex amplitudes for each harmonic of the relevant fields under the QSA. Section \\ref{sec:algorithm} provides details of how the algorithm is implemented. First, the entire numerical workflow that utilizes the three-layer nested loop is described. Next, we introduce the FD implementation of Poisson solvers for each harmonic amplitude and the boundary conditions associated with them. This is followed by a description of the deposition schemes for the source terms for each harmonic needed for the field equations. In Section \\ref{sec:simulation}, we compare simulation results between QPAD, QuickPIC and OSIRIS for the beam-driven wakefields and for the hosing instability. A qualitative discussion on the computational speedup is presented in Section \\ref{sec:performance}. Lastly, we give a conclusion and a discussion for future work.\n\n\\section{Azimuthal decomposition of electromagnetic fields under QSA}\n\\label{sec:theory}\n\nIn this section, we describe the physics arguments behind QPAD including a detailed description of the field equations. As mentioned above, the fundamental differences between a fully explicit 3D PIC code and QPAD are twofold. First, QPAD is a code based on the QSA which separates the time scale of the plasma evolution from that of a drive laser pulse or high-energy particle beam that moves at the speed of light $c$. The assumptions behind the QSA are based on the fact that the characteristic evolution time for a laser driver or a particle beam driver\nis several orders of magnitude larger than the plasma oscillation period, $2\\pi\/\\omega_p$ where $\\omega_p$ is the plasma frequency. In a quasi-static code, a Galilean spatial transformation is made from $(x,y,z,t)$ (where the laser or beam moves in the $\\hat{z}$ direction) to a co-moving frame described by coordinates $(x,y,\\xi=ct-z,s=z)$. All the Lagrangian quantities associated with the plasma particles evolve on the fast-varying time-like variable, $\\xi$, while those of the beam particles moving close to $c$ evolve on the slow-varying \"time\" scale, $s$. The transformations $\\partial_t=c\\partial_\\xi,\\ \\partial_z=\\partial_s-\\partial_\\xi$ are applied for all the Eulerian quantities, \\emph{i.e.}, fields, charge density and current density. The QSA assumes that $s$ is the slow-varying time-like scale, \\emph{i.e.} $\\partial_s\\ll\\partial_\\xi$, so that all the terms associated with $\\partial_s$ are small and can thus be neglected.\n\nFor remainder of the paper, we use normalized units for all the physical quantities; time, length and mass are normalized to $\\omega_p^{-1}$, $c\/\\omega_p$ and the electron rest mass $m_e$. The normalized Maxwell's equations under the QSA can thus be written as\n\\begin{align}\n\\label{eq:maxwell_1}\n\\nabla_\\perp\\times \\bm{E}_\\perp &=-\\PP{B_z}{\\xi}\\bm{e}_z, \\\\\n\\label{eq:maxwell_2}\n\\nabla_\\perp\\times E_z\\bm{e}_z &= -\\PP{}{\\xi}(\\bm{B}_\\perp-\\bm{e}_z\\times\\bm{E}_\\perp), \\\\\n\\label{eq:maxwell_3}\n\\nabla_\\perp\\times \\bm{B}_\\perp - J_z\\bm{e}_z &= \\PP{E_z}{\\xi}\\bm{e}_z, \\\\\n\\label{eq:maxwell_4}\n\\nabla_\\perp\\times B_z\\bm{e}_z - \\bm{J}_\\perp &= \\PP{}{\\xi}(\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp), \\\\\n\\label{eq:maxwell_5}\n\\nabla_\\perp\\cdot\\bm{E}_\\perp-\\rho &= \\PP{E_z}{\\xi}, \\\\\n\\label{eq:maxwell_6}\n\\nabla_\\perp\\cdot\\bm{B}_\\perp &= \\PP{B_z}{\\xi},\n\\end{align}\nwhere $\\nabla_\\perp=\\bm{e}_x\\partial_x+\\bm{e}_y\\partial_y$. For convenience, the equations for the transverse and longitudinal fields are written separately. In this context, transverse and longitudinal are defined with respect to the direction of laser or particle beam propagation and not to the direction of the wavenumber of the fields. Taking linear combinations of Eqs. (\\ref{eq:maxwell_1}), (\\ref{eq:maxwell_3}), (\\ref{eq:maxwell_5}) and (\\ref{eq:maxwell_6}) leads to equations for the divergence and curl of the transverse force, $\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp$, on a particle moving at the speed of light along $\\hat{z}$,\n\\begin{align}\n\\nabla_\\perp\\times(\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp)&=\\bm{0}, \\nonumber \\\\\n\\nabla_\\perp\\cdot(\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp)&=\\rho-J_z. \\nonumber\n\\end{align}\nWe can infer from the first of these equations  that the transverse force can be described by the transverse gradient of a scalar potential which we call $\\psi$,\n\\begin{equation}\n\\label{eq:def_psi}\n\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp=-\\nabla_\\perp\\psi.\n\\end{equation}\nSubstituting this relationship into the second equation, leads to a Poisson equation for the pseudo potential $\\psi$,\n\\begin{equation}\n\\label{eq:poisson_psi}\n-\\nabla_\\perp^2\\psi=(\\rho-J_z).\n\\end{equation}\nBy taking $\\bm{e}_z \\times$ on both sides of Eq. (\\ref{eq:maxwell_2}) and using the relation (\\ref{eq:def_psi}), it can be inferred that $E_z=\\frac{\\partial\\psi}{\\partial\\xi}$. This relationship also follows directly from the definition, $E_z=-\\frac{\\partial \\varphi}{\\partial z}-\\frac{\\partial A_z}{\\partial t}$, and the QSA, where $\\varphi$ and $A_z$ are the scalar potential and the $\\hat{z}$-component of the vector potential.\n\nThe transverse force $\\bm{E}_\\perp+\\bm{e}_z\\times\\bm{B}_\\perp$ in Eq. (\\ref{eq:maxwell_4}) and the quantity $\\bm{B}_\\perp-\\bm{e}_z\\times\\bm{E}_\\perp$ in Eq. (\\ref{eq:maxwell_2}) are not independent. Therefore, the quasi-static form of Maxwell's equations given above cannot be used to advance the fields forward in time, \\emph{i.e.}, $\\xi$, using the FDTD methods as is done in fully explicit PIC codes. Therefore, in QuickPIC, a set of Poisson-like equations are employed to directly solve the fields,\n\\begin{align}\n\\label{eq:poisson_bxy}\n\\nabla_\\perp^2\\bm{B}_\\perp&=\\bm{e}_z\\times\\left(\\PP{\\bm{J}_\\perp}{\\xi}+\\nabla_\\perp J_z\\right), \\\\\n\\label{eq:poisson_bz}\n\\nabla_\\perp^2 B_z&=-\\bm{e}_z\\cdot(\\nabla_\\perp\\times\\bm{J}_\\perp), \\\\\n\\label{eq:poisson_ez}\n\\nabla_\\perp^2 E_z&=\\nabla_\\perp\\cdot\\bm{J}_\\perp.\n\\end{align}\nwhich can be derived by applying the QSA to the wave equations for $\\bm{E}$ and $\\bm{B}$. After obtaining $\\bm{B}_\\perp$ from Eq. (\\ref{eq:poisson_bxy}) and $\\psi$ from Eq. (\\ref{eq:poisson_psi}), we can calculate $\\bm{E}_\\perp$ by subtracting $\\bm{e}_z\\times\\bm{B}_\\perp$ from $-\\nabla_\\perp\\psi$. Although it is not directly used in QuickPIC, for completeness we write out the Poisson-like equation for $\\bm{E}_\\perp$,\n\\begin{equation}\n\\label{eq:poisson_exy}\n\\nabla_\\perp^2\\bm{E}_\\perp=\\nabla_\\perp\\rho+\\PP{\\bm{J}_\\perp}{\\xi}.\n\\end{equation}\n\nWe next expand the electromagnetic fields, charge density and current density in cylindrical coordinates with each quantity being decomposed into a Fourier series in the azimuthal direction. To obtain a set of equations for the Fourier amplitude of each azimuthal harmonic, we first write the field equations, Eqs. (\\ref{eq:def_psi})-(\\ref{eq:poisson_exy}), in cylindrical coordinates,\n\\begin{align}\n\\label{eq:fperp}\n\\bm{e}_r\\PP{\\psi}{r}+\\bm{e}_\\phi\\frac{1}{r}\\PP{\\psi}{\\phi}&=(-E_r+B_\\phi)\\bm{e}_r-(E_\\phi+B_r)\\bm{e}_\\phi, \\\\\n\\nabla_\\perp^2\\psi&=-(\\rho-J_z), \\\\\n\\label{eq:pois_br}\n\\nabla_\\perp^2B_r-\\frac{B_r}{r^2}-\\frac{2}{r^2}\\PP{B_\\phi}{\\phi}&=-\\PP{J_\\phi}{\\xi}-\\frac{1}{r}\\PP{J_z}{\\phi}, \\\\\n\\nabla_\\perp^2B_\\phi-\\frac{B_\\phi}{r^2}+\\frac{2}{r^2}\\PP{B_r}{\\phi}&=\\PP{J_r}{\\xi}+\\PP{J_z}{r}, \\\\\n\\nabla_\\perp^2B_z&=-\\frac{1}{r}\\PP{}{r}(rJ_\\phi)+\\frac{1}{r}\\PP{J_r}{\\phi}, \\\\\n\\nabla_\\perp^2E_r-\\frac{E_r}{r^2}-\\frac{2}{r^2}\\PP{E_\\phi}{\\phi}&=\\PP{\\rho}{r}+\\PP{J_r}{\\xi}, \\\\\n\\nabla_\\perp^2E_\\phi-\\frac{E_\\phi}{r^2}+\\frac{2}{r^2}\\PP{E_r}{\\phi}&=\\frac{1}{r}\\PP{\\rho}{\\phi}+\\PP{J_\\phi}{\\xi}, \\\\\n\\label{eq:pois_ez}\n\\nabla_\\perp^2E_z&=\\frac{1}{r}\\PP{}{r}(rJ_r)+\\frac{1}{r}\\PP{J_\\phi}{\\phi}.\n\\end{align}\nwhere the 2D (transverse) Laplacian is defined as $\\nabla_\\perp^2\\equiv\\frac{1}{r}\\PP{}{r}\\left(r\\PP{}{r}\\right)+\\frac{1}{r}\\PP{^2}{\\phi^2}$. Expanding the electromagnetic fields, charge and current density into a Fourier series in the azimuthal direction, gives\n\\begin{equation}\n\\label{eq:fourier_series}\n\\begin{aligned}\nU(r,\\phi)&=\\sum_{m=-\\infty}^{+\\infty}U^m(r)e^{\\mathrm{i} m\\phi}\\\\\n&=U^0(r)+2\\sum_{m=1}\\mathfrak{Re}\\{U^m\\}\\cos(m\\phi)-2\\sum_{m=1}\\mathfrak{Im}\\{U^m\\}\\sin(m\\phi)\n\\end{aligned}\n\\end{equation}\nwhere $U$ represents an arbitrary scalar field or components of a vector field, and note that the amplitude of each harmonic $U^m$ is complex. It follows that $U^{-m}=(U^m)^*$ because $U(r,\\phi)$ is real, which indicates that only the evolution of $m\\ge0$ modes need to be considered. Substituting the expansion into Eqs. (\\ref{eq:fperp})-(\\ref{eq:pois_ez}) yields the following governing equations for each mode\n\\begin{align}\n\\label{eq:fperp_m}\n\\bm{e}_r\\PP{\\psi^m}{r}+\\bm{e}_\\phi\\frac{\\mathrm{i} m}{r}\\psi^m&=(-E_r^m+B_\\phi^m)\\bm{e}_r-(E_\\phi^m+B_r^m)\\bm{e}_\\phi,\\\\\n\\label{eq:pois_psi_m}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m \\psi^m&=-(\\rho^m-J_z^m), \\\\\n\\label{eq:pois_br_m}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_r^m-\\frac{B_r^m}{r^2}-\\frac{2\\mathrm{i} m}{r^2}B_\\phi^m&=-\\PP{J_\\phi^m}{\\xi}-\\frac{\\mathrm{i} m}{r}J_z^m, \\\\\n\\label{eq:pois_bphi_m}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_\\phi^m-\\frac{B_\\phi^m}{r^2}+\\frac{2\\mathrm{i} m}{r^2}B_r^m&=\\PP{J_r^m}{\\xi}+\\PP{J_z^m}{r}, \\\\\n\\label{eq:pois_bz_m}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_z^m&=-\\frac{1}{r}\\PP{}{r}(rJ_\\phi^m)+\\frac{\\mathrm{i} m}{r}J_r^m, \\\\\n\\mathop{}\\!\\mathbin\\bigtriangleup_m E_r^m-\\frac{E_r^m}{r^2}-\\frac{2\\mathrm{i} m}{r^2}E_\\phi^m&=\\PP{\\rho^m}{r}+\\PP{J_r^m}{\\xi}, \\\\\n\\mathop{}\\!\\mathbin\\bigtriangleup_m E_\\phi^m-\\frac{E_\\phi^m}{r^2}+\\frac{2\\mathrm{i} m}{r^2}E_r^m&=\\frac{\\mathrm{i} m}{r}\\rho^m+\\PP{J_\\phi^m}{\\xi}, \\\\\n\\label{eq:pois_ez_m}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m E_z^m&=\\frac{1}{r}\\PP{}{r}(rJ_r^m)+\\frac{\\mathrm{i} m}{r}J_\\phi^m\n\\end{align}\nwhere $\\mathop{}\\!\\mathbin\\bigtriangleup_m\\equiv\\frac{1}{r}\\PP{}{r}\\left(r\\PP{}{r}\\right)-\\frac{m^2}{r^2}$. This set of equations is overdetermined and therefore, similarly to what is currently used in the 3D QuickPIC algorithm \\cite{an2013}, we select Eqs. (\\ref{eq:fperp_m})-(\\ref{eq:pois_bz_m}) and (\\ref{eq:pois_ez_m}) to solve for the electromagnetic fields.\n\nSimilar to other QSA codes and Darwin model codes \\cite{nielson1976}, it is not straightforward to solve the Poisson-like equations and therefore a predictor-corrector iteration is necessary to implicitly determine part of field components. The difficulty in our code arises because the source terms in Eqs. (\\ref{eq:pois_br_m}) and (\\ref{eq:pois_bphi_m}) are not known at the appropriate time step.  We use the same time indexing as in QuickPIC \\cite{huang2006,an2013}. The momentum $\\bm{p}$ and Lorentz factor $\\gamma$ for the plasma particles are defined on integer time steps, $\\xi=n_\\xi\\Delta_\\xi$, while the transverse position $\\bm{x}_\\perp$ and all the Eulerian quantities including $\\psi^m$, $\\bm{E}^m$, $\\bm{B}^m$, $(\\rho-J_z)^m$, $\\bm{J}^m$ and $\\pp{\\bm{J}_\\perp^m}{\\xi}$ are defined on half-integer time steps, $\\xi=(n_\\xi+\\frac{1}{2})\\Delta_\\xi$. In order to deposit $\\pp{\\bm{J}_\\perp^m}{\\xi}$ and $\\bm{J}^m$, the momentum $\\bm{p}^{n_\\xi+\\frac{1}{2}}$ (the superscript denotes the index of $\\xi$) needs to be known. These could be obtained by averaging  $\\bm{p}^{n_\\xi+1}$ and $\\bm{p}^{n_\\xi}$ but $\\bm{p}^{n_\\xi+1}$ is not known because the fields at $\\xi=(n_\\xi+\\frac{1}{2})\\Delta_\\xi$  are not known.  Therefore, an iteration procedure is needed.\nThe $\\bm{B}^m$ and $\\bm{E}^m$ solved at $\\xi=(n_\\xi-\\frac{1}{2})\\Delta_\\xi$ are used as an appropriate initial guess at $\\xi=(n_\\xi+\\frac{1}{2})\\Delta_\\xi$.\nThese are then used to predict $\\bm{p}^{n_\\xi+1}$ in a leapfrog manner and the $\\bm{p}^{n_\\xi+\\frac{1}{2}}$ is simply evaluated by the average $(\\bm{p}^{n_\\xi}+\\bm{p}^{n_\\xi+1})\/2$, which we call the predictor procedure. We note that as described in ref. \\cite{an2013}, $\\pp{\\bm{J}_\\perp^m}{\\xi}$ is obtained by analytically evaluating the derivative of the shape function and not through a finite difference operation of $\\bm{J}_\\perp^m$. Using this method the particle positions do not need be updated within the predictor procedure. The predicted $\\bm{p}^{n_\\xi+\\frac{1}{2}}$ are then used to deposit the source terms $\\pp{\\bm{J}_\\perp^m}{\\xi}$ and $\\bm{J}^m$  which are used to  improve  the values of $\\bm{B}^m$ and $\\bm{E}^m$ from the initial guesses\/predictions. This operation is called corrector procedure. To guarantee the procedure for correcting $\\bm{B}^m_\\perp$ is stable and that it converges, an iterative form of the Poisson equation is used\n\\begin{align}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_r^{m,l+1}-\\left(1+\\frac{1}{r^2}\\right)B_r^{m,l+1}-\\frac{2\\mathrm{i} m}{r^2}B_\\phi^{m,l+1}&=-\\left(\\PP{J_\\phi^m}{\\xi}\\right)^l-\\frac{\\mathrm{i} m}{r}J_z^{m,l}-B_r^{m,l}, \\nonumber \\\\\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_\\phi^{m,l+1}-\\left(1+\\frac{1}{r^2}\\right)B_\\phi^{m,l+1}+\\frac{2\\mathrm{i} m}{r^2}B_r^{m,l+1}&=\\left(\\PP{J_r^m}{\\xi}\\right)^l+\\PP{J_z^{m,l}}{r}-B_\\phi^{m,l}, \\nonumber\n\\end{align}\nwhere the superscript $l$ denotes the iteration step. The other components of the fields, $\\bm{E}_\\perp^m$, $B_z^m$, $E_z^m$ can then be obtained once $\\bm{B}_\\perp$ is known via Eqs. (\\ref{eq:fperp_m}), (\\ref{eq:pois_bz_m}) and ($\\ref{eq:pois_ez_m}$) respectively (note that $\\psi^m$ is already known before the predictor-corrector iteration). This predictor-corrector iteration can be conducted for an arbitrary number of times until the answers are convergence to a desired accuracy.\n\nUnlike in 3D QuickPIC where the equations for the two components of the $\\bm{B}_\\perp$ are decoupled, Eqs. (\\ref{eq:pois_br_m}) and (\\ref{eq:pois_bphi_m}) are coupled. For numerical reasons, we instead seek solutions to a set of decoupled equations by introducing new variables $B_+^m=B_r^m+\\mathrm{i} B_\\phi^m$ and $B_-^m=B_r^m-\\mathrm{i} B_\\phi^m$ in QPAD. With these new field variables, the decoupled equations can be written as\n\\begin{equation}\n\\label{eq:pois_decouple_m_iter}\n\\left(\\PP{^2}{r^2}+\\frac{1}{r}\\PP{}{r}-\\frac{(m\\pm1)^2}{r^2}-1\\right)B_\\pm^{m,l+1}=S_\\pm^{m,l}-B_\\pm^{m,l}\n\\end{equation}\nwhere\n\\begin{equation}\nS_\\pm^m=-\\PP{J_\\phi^m}{\\xi}-\\frac{\\mathrm{i} m}{r}J_z^m\\pm\\mathrm{i}\\left(\\PP{J_r^m}{\\xi}+\\PP{J_z^m}{r}\\right). \\nonumber\n\\end{equation}\nAs we will see in the next section, after discretization, the decoupled equations become tri-diagonal linear systems for which the efficient cyclic reduction algorithm \\cite{press2010} can be applied. On the other hand, the original coupled equations would be solved using classic iterative methods or sparse matrix techniques which typically are computationally less efficient.\n\nFor computationally simplicity, in the azimuthal mode expansion method, we treat the fields from the beam separately. Due to the approximation that the transverse current $\\bm{J}_\\perp$ is negligible for beam particles and these particles travel at a speed very closed to the speed of light, $c$ it follows that $\\rho^m_\\text{beam}\\simeq J_{z,\\text{beam}}^m$. There is thus no transverse current from the beam which implies that longitudinal fields $B_z^m$ and $E_z^m$ from the beam vanish, and that Eqs. (\\ref{eq:pois_br_m}) and (\\ref{eq:pois_bphi_m}) reduce to an electrostatics problem,\n\\begin{equation}\n\\bm{B}^m_{\\perp,\\text{beam}}=\\bm{e}_r\\frac{\\mathrm{i} m}{r}A_z^m+\\bm{e}_\\phi\\PP{A_z^m}{r} \\nonumber\n\\end{equation}\nand $A_z^m$ satisfies\n\\begin{equation}\n\\label{eq:pois_phi_m}\n-\\mathop{}\\!\\mathbin\\bigtriangleup_m A^m_z=J^m_{z,\\text{beam}}=\\rho^m_\\text{beam}.\n\\end{equation}\nOnce  $\\bm{B}^m_{\\perp,\\text{beam}}$ is known then the electric fields can be obtained through $E^m_{r,\\text{beam}}=B^m_{\\phi,\\text{beam}}$ and $E^m_{\\phi,\\text{beam}}=-B^m_{r,\\text{beam}}$.\n\n\\section{Algorithm implementation}\n\\label{sec:algorithm}\n\n\\subsection{Numerical workflow in QPAD}\n\\label{subsec:workflow}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{workflow.pdf}\n\\caption{The numerical workflow of QPAD}\n\\label{fig:workflow}\n\\end{figure}\n\nIn this section, we briefly introduce the numerical workflow in QPAD.  QPAD consists of three loops (see figure \\ref{fig:workflow}). The outermost level is the the quasi-3D loop in which the charge (current) of the beam particles are deposited onto the $r$-$\\xi$ plane for multiple Fourier harmonics, and the beam particles are pushed in $s$ in the full 3D space described by $(x,y,\\xi)$ coordinates. The particles are pushed using the  leapfrog method with second-order accuracy.\n\nThe quasi-2D loop is embedded into the quasi-3D loop to solve the harmonic amplitudes for all the fields with the plasma and beam charges and currents as sources. The motion of plasma particles is in the 2D space described by $(x, y)$ and particles are pushed in the coordinate $\\xi$. In this loop, the evolution of fields and the motion of plasma particles are updated slice by slice along the negative $\\xi$-direction. The transverse fields from the particle beam are first calculated at a given slice. This together with the self-consistent fields from the plasma particles are used to advance the particles to new position and momenta at the next slice. In the quasi-static algorithm the particle's charge depends on its speed in the $\\hat z$ direction and there are well defined relationships between $\\bm p_z$, $\\bm p_\\perp$ and $\\psi$. Therefore, the pseudo-potential $\\psi$ must also be interpolated to each particle's position and stored for the subsequent particle push. The equation of motion for a plasma particle is,\n\\begin{equation}\n\\DD{\\bm{p}_\\perp}{\\xi}=\\frac{q\\gamma}{1+\\frac{q}{m}\\psi}\\left[\\bm{E}_\\perp+\\left(\\frac{\\bm{p}}{\\gamma}\\times\\bm{B}\\right)_\\perp\\right] \\nonumber\n\\end{equation}\nand\n\\begin{equation}\np_z=\\frac{1+p_\\perp^2-(1-\\frac{q}{m}\\psi)^2}{2(1-\\frac{q}{m}\\psi)}. \\nonumber\n\\end{equation}\nOnce $\\psi$ is known, then the transverse fields $E_r-B_\\phi$ and $E_\\phi+B_r$ can be obtained by taking a transverse gradient of $\\psi$ according to Eq. (\\ref{eq:fperp_m}). The next step is to call the predictor-corrector iteration to implicitly solve the fields induced by plasma as described earlier. The iteration loop starts with updating the particle momenta by using an initial guess for $\\bm{E}$ and $\\bm{B}$. The predicted momenta are then used to deposit the source terms $\\bm{J}$ and $\\pp{\\bm{J}_\\perp}{\\xi}$ needed to solve for $\\bm{B}_\\perp$. The updated $\\bm{E}_\\perp$ is evaluated by subtracting $\\bm{B}_\\perp$ from $-\\nabla_\\perp\\psi$ according to Eq. (\\ref{eq:fperp_m}). With the updated $\\bm{J}$, the longitudinal field components $E_z$ and $B_z$ can be straightforwardly solved using Eqs. (\\ref{eq:pois_ez_m}) and (\\ref{eq:pois_bz_m}). This iteration is terminated when a maximum iterative step is reached or the updated fields meet a specified criterion for convergence\n\\begin{equation}\n\\frac{\\max|\\bm{B}^{l+1}-\\bm{B}^l|}{\\max|\\bm{B}^l|}<tol. \\nonumber\n\\end{equation}\nwhere the chosen tolerance is typically $10^{-3}$ or smaller. The last step in the quasi-2D loop is pushing the plasma particles with the converged electromagnetic fields (interpolating the fields onto the particle position) and $\\psi$ previously stored on each particle.\n\nIn QPAD, the position and momenta of plasma particles are advanced using the classic Boris integrator \\cite{boris1972,qin2013} in a leapfrog manner. All the fields in QPAD are solved using  second-order accurate finite difference methods in conjunction with the multi-grid method. A finite difference version of free boundary conditions for different types of field components are implemented into QPAD as well. New current and charge deposition schemes needed for cylindrical geometry and azimuthal decomposition are also developed and implemented. In the next two sections, we describe the numerical implementation of the field solver and deposition in detail.\n\n\n\\subsection{Finite difference Poisson field solver}\n\nIn 3D quasi-static PIC codes based on Cartesian coordinates, e.g., QuickPIC and HiPACE \\cite{mehrling2014} the fields are solved using FFTs as they are fast and accurate. High parallel scalability is obtained through careful considerations on minimizing data transfer and the use of a pipelining algorithm \\cite{feng2009,an2013}. In QPAD, we adopt finite difference (FD) methods to solve Poisson equations because FFTs can no longer be directly used in cylindrical geometry. We define all the fields on the integer grid points $r_i$, \\emph{i.e.} $r_i=i\\Delta_r$ for $i=1,\\dots,N$ where $N$ is the total number of grid points in $\\hat{r}$-direction and $\\Delta_r$ is the radial cell size. Using a 3-point discretization, the $\\mathop{}\\!\\mathbin\\bigtriangleup_m$ operator with second-order precision can be written as\n\\begin{equation}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m U\\rightarrow\n\\beta_i^- U_{i-1}-\\alpha_i U_i+\\beta_i^+ U_{i+1} \\nonumber\n\\end{equation}\nwhere\n\\begin{equation}\n\\beta_i^\\pm=\\frac{1}{\\Delta_r^2}\\pm\\frac{1}{2r_i\\Delta_r},\\quad\n\\alpha_i=\\frac{2}{\\Delta_r^2}+\\frac{m^2}{r_i^2} \\nonumber\n\\end{equation}\nand $U$ is an arbitrary scalar field. The operator $\\pp{}{r}$ is approximated with the a central difference indexing with second-order precision. Similarly, the operator in Eq. (\\ref{eq:pois_decouple_m_iter}) is discretized as\n\\begin{equation}\n\\left(\\PP{^2}{r^2}+\\frac{1}{r}\\PP{}{r}-\\frac{(m\\pm1)^2}{r^2}-1\\right)U\\rightarrow\n\\beta_i^- U_{i-1}- \\mu_i U_i+\\beta_i^+ U_{i+1} \\nonumber\n\\end{equation}\nwith\n\\begin{equation}\n\\mu_i=\\frac{2}{\\Delta_r^2}+\\frac{(m\\pm1)^2}{r_i^2}+1. \\nonumber\n\\end{equation}\n\nIn QPAD, the governing Eqs. (\\ref{eq:fperp_m}), (\\ref{eq:pois_psi_m}), (\\ref{eq:pois_bz_m}), (\\ref{eq:pois_ez_m}), (\\ref{eq:pois_decouple_m_iter}) and (\\ref{eq:pois_phi_m}) are all discretized. These Poisson equations are converted into tri-diagonal linear systems which can benefit from fast solvers using the cyclic reduction method. These solvers are implemented with the library Hypre \\cite{falgout2002} developed and maintained by LLNL.\n\n\n\\subsection{Boundary conditions}\n\nBoth conducting and free (open) boundary conditions have been implemented in QPAD. Conducting boundary conditions are more standard and are not described further. The basic idea for free or open boundaries is to assume that the space outside the computational domain is vacuum and that it extends to infinity. Therefore, solutions can be obtained by solving a series of Laplace equations. The boundary values can be determined by utilizing the fact that the fields are continuous at the boundary. See \\ref{sec:app:free_bnd} for the details of the derivation.\n\nWhen implementing the field solvers in cylindrical geometry, issues with respect to singularities on the axis are inevitable issue because of the presence of the $1\/r$ term. As we discussed above, all the field components in QPAD are defined on integer grid points. Therefore,  all of the Poisson equations can have a singularity at $r=0$. These singularities can be properly treated by considering the properties of different field components at $r=0$. As pointed out by Constantinescu and Lele \\cite{constantinescu2002}, for any scalar of component of a field in a Cartesian coordinate directions, \\emph{i.e.}, $(\\psi, \\varphi, E_z, B_z, \\rho-J_z, J_z)$, the $m=0$ mode is non-zero at $r=0$ while other modes are zero at $r=0$.\nOn the other hand, for the field components in cylindrical coordinate directions $(E_r, E_\\phi, B_r, B_\\phi, J_r, J_\\phi)$ the $m=1$ mode is non-zero at $r=0$ and the other modes are zero at $r=0$.\n\nThe field components whose boundary values at $r=0$ need to be determined therefore only include $\\psi^0$, $\\varphi^0$, $B_z^0$, $E_z^0$ and $B_\\pm^1$; all other field components vanish at $r=0$. The singularity of the  $\\frac{1}{r}\\PP{U}{r}$ term on the LHS of each Poisson equation (where $U$ denotes any of fields mentioned above) can be eliminated by applying L'Hospital's rule, so that $\\frac{1}{r}\\PP{U}{r}\\rightarrow\\PP{^2U}{r^2}$. The terms having $1\/r$ on the RHS of Poisson equation can be treated in the same manner. There is another important property for components in the cylindrical coordinate directions \\cite{constantinescu2002}. The combinations $U_r+imU_\\phi$ and $U_\\phi-imU_r$ vanish at $r=0$ for arbitrary $m$, which implies $B^1_+$ (recalling the definition is $B_+^m\\equiv B_r^m+iB_\\phi^m$) vanishes at $r=0$ even though both $B_r^1$ and $B_\\phi^1$ are non-zero on the axis. For $m\\neq1$ modes $B_r^m$ and $B_\\phi^m$ are naturally zero at $r=0$ according to previous discussion, therefore, $B_+^m$ vanishes on the axis for arbitrary $m$. Considering the symmetry of different fields around the axis, the discrete version of boundary conditions at $r=0$ can therefore be summarized as follows,\n\\begin{align}\n4(\\psi_1^{m=0}-\\psi_0^{m=0}) &= -(\\rho-J_z)_0^{m=0} \\Delta_r^2 \\\\\n2(E_{z,1}^{m=0}-E_{z,0}^{m=0}) &= J_{r,1}^{m=0} \\Delta_r \\\\\n2(B_{z,1}^{m=0}-B_{z,0}^{m=0}) &= -J_{\\phi,1}^{m=0} \\Delta_r \\\\\n4(B_{-,1}^{m=1}-B_{-,0}^{m=1}) &= -\\left[\\left(\\PP{J_\\phi}{\\xi}\\right)_0^{m=1} + i\\left(\\PP{J_r}{\\xi}\\right)_0^{m=1} + 2i\\frac{J_{z,1}^{m=1}}{\\Delta_r} \\right] \\Delta_r^2\n\\end{align}\n\n\\subsection{Deposition of source terms}\n\nIn order to solve the 1D Poisson equations for each harmonic amplitude, the source terms on the RHS of the governing Eqs. (\\ref{eq:fperp_m})-(\\ref{eq:pois_bz_m}) and (\\ref{eq:pois_ez_m}) must be deposited from the particle information (charge, position and momentum) onto the grid points. The source terms to be deposited include $\\rho^m-J_z^m$, $\\bm{J}^m$ and $\\pp{\\bm{J}_{\\perp}^m}{\\xi}$. Since these source terms are defined on the grid in the $\\hat{r}$-direction while the particles are described by Cartesian coordinates, we need to transform the particle positions and momenta from the cylindrical to Cartesian coordinates in QPAD. The following equation is used to deposit the current as in QuickPIC,\n\\begin{equation}\n\\label{eq:dep_jay}\n\\bm{J}=\\frac{1}{\\text{Vol.}}\\sum_i\\frac{q_i\\bm{v}_i}{1-v_{iz}}S(\\bm{x}_\\perp-\\bm{x}_{i\\perp})=\\frac{1}{\\text{Vol.}}\\sum_i\\frac{q_i\\bm{p}_i}{1-\\frac{q_i}{m_i}\\psi_i}S(\\bm{x}_\\perp-\\bm{x}_{i\\perp})\n\\end{equation}\nwhere $S(\\bm{x}_\\perp-\\bm{x}_{i\\perp})$ is the particle shape function to interpolate the particle quantities at $i$th particle's transverse position $\\bm{x}_{i\\perp}$ onto the grid position $\\bm{x}_\\perp$. The pseudo-potential felt by an individual particle $\\psi_i$ is obtained by interpolating the $\\psi$ solved on the grid to the position of the particle. The second expression for $\\bm J$ can be obtained by multiplying the numerator and denominator of the first expression by the Lorentz factor $\\gamma_i$, and using the constant of motion under the QSA, $\\gamma-p_z=1-(q\/m)\\psi$. In order to derive the deposition scheme in QPAD in which the azimuthal direction is gridless, we  expand $S(\\bm{x}_\\perp-\\bm{x}_{i\\perp})$ into a Fourier series as well. In cylindrical geometry, the interpolation function is defined as\n\\begin{equation}\nS(\\bm{x}_\\perp-\\bm{x}_{i\\perp})\\equiv\n\\frac{1}{r}S_r(r-r_i)S_\\phi(\\phi-\\phi_i), \\nonumber\n\\end{equation}\nwhich is subject to the normalization condition $\\int dr d\\phi\\ S_rS_\\phi=1$. Next, we expand $S_\\phi$ into azimuthal harmonics\n\\begin{equation}\nS_\\phi(\\phi-\\phi_i)=\\sum_m S_\\phi^m(\\phi_i)e^{\\mathrm{i} m\\phi} \\nonumber\n\\end{equation}\nwhere\n\\begin{equation}\nS_\\phi^m(\\phi_i)=\\frac{1}{2\\pi}\\int_0^{2\\pi}d\\phi'\\ S_\\phi(\\phi'-\\phi_i)e^{-\\mathrm{i} m\\phi'}, \\nonumber\n\\end{equation}\nand require both $S_r$ and $S_\\phi$ to satisfy the normalization condition $\\int drS_r=1$ and $\\int d\\phi S_\\phi=1$. If we take $S_\\phi$ to be a Dirac delta function(which we do in QPAD), then $S_\\phi^m=\\frac{1}{2\\pi}e^{-\\mathrm{i} m\\phi_i}$. In addition,  the current $\\bm{J}$ defined on the {\\it r-z} grid can be expanded as\n\\begin{equation}\n\\bm{J}(r,\\phi)=\\sum_m \\bm{J}^m(r) e^{\\mathrm{i} m\\phi}, \\nonumber\n\\end{equation}\nthe deposition for $\\bm{J}^m$ is found to be\n\\begin{equation}\n\\bm{J}^m=\\frac{1}{\\text{Vol.}}\\sum_i\\frac{q_i\\bm{p}_i}{1-\\frac{q_i}{m_i}\\psi_i}\\frac{1}{r}S_r(r-r_i)S^m_\\phi(\\phi_i). \\nonumber\n\\end{equation}\nTherefore, it is actually not necessary to calculate each {\\it m} mode but only the $m=0$ mode from each particle. Any $m>0$ mode for an individual particle can be obtained from the $m=0$ contribution by simply multiplying by a phase factor through the relation $\\bm{J}^m=\\bm{J}^0e^{-\\mathrm{i} m\\phi_i}$ or recursively through $\\bm{J}^m=\\bm{J}^{m-1}e^{-\\mathrm{i}\\phi_i}$ if $S(\\phi-\\phi_i)=\\delta(\\phi-\\phi_i)$ is used.\n\nLikewise, according to ref. \\cite{an2013}, the deposition for $(\\rho-J_z)^m$ can be written as\n\\begin{equation}\n(\\rho-J_z)^m=\\frac{1}{\\text{Vol.}}\\sum_i\\frac{q_i}{r}S_r(r-r_i)S_\\phi^m(\\phi_i), \\nonumber\n\\end{equation}\nwhere $(\\rho-J_z)^m=(\\rho-J_z)^{m-1}e^{-\\mathrm{i}\\phi_i}$ for each particle.\n\nIn section \\ref{subsec:workflow}, we showed that in the predictor-corrector iteration the source term $\\pp{\\bm{J}^m_\\perp}{\\xi}$ at the half-integer time step $\\xi=(n+1\/2)\\Delta_\\xi$ needs to be calculated. This can be done in two ways. The first method, which was adopted in the original version of QuickPIC \\cite{huang2006}, is to predict $J^m_r$ and $J_\\phi^m$ at the next integer time step $\\xi=(n_\\xi+1)\\Delta_\\xi$ and approximate the derivative using the centered difference $\\pp{J_{r,\\phi}^m}{\\xi}\\vert^{n_\\xi+\\frac{1}{2}}=(J_{r,\\phi}^m\\vert^{n_\\xi+1}-J_{r,\\phi}^m\\vert^{n_\\xi})\/\\Delta_\\xi$. However, this approach requires repartitioning the particles within a single pass through the iteration loop when using domain decomposition as it requires updating the particle positions and storing previous and predicted values. In the current version of QuickPIC \\cite{an2013}, this approach is replaced by analytically calculating the derivative of the current in terms of $\\bm{x}_\\perp$, $\\bm{p}_\\perp$ and $\\psi$ using their particle shapes, which allows direct deposition without the computationally expensive particle repartitioning procedure. In QPAD, we use the approach in the current version of  QuickPIC to deposit $\\pp{\\bm{J}_\\perp^m}{\\xi}$. By definition, we have\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:dep_djrdxi}\n\\PP{J_r^m}{\\xi}&=\\frac{1}{\\text{Vol.}}\\sum_i \\PP{}{\\xi}\\left(\\frac{q_ip_{r,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\frac{1}{r}S_rS_\\phi^m\\right) \\\\\n&=\\frac{1}{\\text{Vol.}}\\sum_i \\frac{q_i}{r}\\left(\\frac{\\dd{p_{r,i}}{\\xi}}{1-\\frac{q_i}{m_i}\\psi_i}S_rS_\\phi^m+\\frac{p_{r,i}\\dd{(\\frac{q}{m_i}\\psi_i)}{\\xi}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}S_rS_\\phi^m+\\frac{p_{r,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\PP{(S_rS_\\phi^m)}{\\xi}\\right).\n\\end{aligned}\n\\end{equation}\nIt should be pointed out that the $\\psi_i$ in the denominator is the total value which is obtained by summing all the harmonics. The derivative of $\\psi_i$ with respect to $\\xi$ is calculated by\n\\begin{equation}\n\\label{eq:dpsidxi}\n\\DD{\\psi_i}{\\xi}=E_{z,i}+\\PP{\\psi_i}{r_i}\\DD{r_i}{\\xi}+\\PP{\\psi_i}{\\phi_i}\\DD{\\phi_i}{\\xi}\n\\end{equation}\nwhere the terms $E_{z,i}$, $\\PP{\\psi_i}{r_i}$ and $\\PP{\\psi_i}{\\phi_i}$ are regarded as the interpolated value of $E_z$, $\\PP{\\psi}{r}$ and $\\PP{\\psi}{\\phi}$ at the particle's position $(r_i,\\phi_i)$. The terms $\\DD{r_i}{\\xi}$ and $\\DD{\\phi_i}{\\xi}$ are evaluated by\n\\begin{equation}\n\\DD{r_i}{\\xi}=\\frac{p_{r,i}}{1-\\frac{q_i}{m_i}\\psi_i},\\quad\n\\DD{\\phi_i}{\\xi}=\\frac{1}{r_i}\\frac{p_{\\phi,i}}{1-\\frac{q_i}{m_i}\\psi_i}. \\nonumber\n\\end{equation}\nFor the last term in the bracket of Eq. (\\ref{eq:dep_djrdxi}), $\\pp{(S_rS_\\phi^m)}{\\xi}$ is calculated by\n\\begin{equation}\n\\begin{aligned}\n\\PP{}{\\xi}\\left(S_r(r-r_i)S_\\phi^m(\\phi_i)\\right)&=-\\DD{r_i}{\\xi}\\PP{S_r}{r}S_\\phi^m+\\DD{\\phi_i}{\\xi}\\PP{S_\\phi^m}{\\phi_i}S_r \\\\\n&=-\\frac{p_{r,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\PP{S_r}{r}S_\\phi^m+\\frac{p_{\\phi,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\frac{1}{r_i}\\PP{S_\\phi^m}{\\phi_i}S_r \\\\\n&=-\\frac{e^{-\\mathrm{i} m\\phi_i}}{2\\pi}\\left(\\frac{p_{r,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\PP{S_r}{r}+\\frac{p_{\\phi,i}}{1-\\frac{q_i}{m_i}\\psi_i}\\frac{\\mathrm{i} mS_r}{r_i}\\right). \\nonumber\n\\end{aligned}\n\\end{equation}\nwhere we have applied $S_\\phi=\\delta(\\phi-\\phi_i)$ again. Substituting these expressions into Eq. (\\ref{eq:dep_djrdxi}), we finally obtain the deposition for $\\pp{J_r^m}{\\xi}$\n\\begin{equation}\n\\begin{aligned}\n\\PP{J_r^m}{\\xi}&=\\frac{1}{2\\pi\\text{Vol.}}\\left\\{\\sum_i q_ie^{-\\mathrm{i} m\\phi_i}\\left(\\frac{\\dd{p_{r,i}}{\\xi}}{1-\\frac{q_i}{m_i}\\psi_i}+\\frac{p_{r,i}\\dd{(\\frac{q}{m}\\psi_i)}{\\xi}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}-\\frac{p_{r,i}p_{\\phi,i}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{\\mathrm{i} m}{r_i}\\right.\\right.\\\\\n&\\left.\\left.-\\frac{p_{r,i}^2}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{1}{r}\\right)\\frac{S_r}{r}-\\PP{}{r}\\left(\\sum_i q_i e^{-\\mathrm{i} m\\phi_i}\\frac{p_{r,i}^2}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{S_r}{r}\\right)\\right\\}\n\\end{aligned}, \\nonumber\n\\end{equation}\nand likewise we can derive the deposition formula for $\\pp{J_\\phi^m}{\\xi}$\n\\begin{equation}\n\\begin{aligned}\n\\PP{J_\\phi^m}{\\xi}&=\\frac{1}{2\\pi\\text{Vol.}}\\left\\{\\sum_i q_ie^{-\\mathrm{i} m\\phi_i}\\left(\\frac{\\dd{p_{\\phi,i}}{\\xi}}{1-\\frac{q_i}{m_i}\\psi_i}+\\frac{p_{\\phi,i}\\dd{(\\frac{q}{m}\\psi_i)}{\\xi}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}-\\frac{p_{\\phi,i}^2}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{\\mathrm{i} m}{r_i}\\right.\\right.\\\\\n&\\left.\\left.-\\frac{p_{r,i}p_{\\phi,i}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{1}{r}\\right)\\frac{S_r}{r}-\\PP{}{r}\\left(\\sum_i q_i e^{-\\mathrm{i} m\\phi_i}\\frac{p_{r,i}p_{\\phi,i}}{(1-\\frac{q_i}{m_i}\\psi_i)^2}\\frac{S_r}{r}\\right)\\right\\}.\n\\end{aligned}\n\\end{equation}\n\n\\section{Simulation results}\n\\label{sec:simulation}\n\nIn this section, we present a small sample of benchmark tests for QPAD compared against results from QuickPIC and 3D OSIRIS. These benchmarks are related to the plasma wakefield accelerator (PWFA) concept which uses high-energy particle beams to excite a plasma wave wake. The plasma wake provides very large accelerating and focusing forces as compared with conventional accelerator structures. These fields can be used to accelerate and\/or focus a trailing beam riding on an appropriate phase inside the wake. We present benchmarks for driving wakefields in both the linear and nonlinear regimes with only a single mode (only $m=0$ mode). We also present a benchmark for a case where a second witness beam is placed inside a nonlinear wakefield \\cite{lu2006,rosenzweig1991} with an offset in one direction with respect to the drive beam. This leads to a hosing instability \\cite{whittum1991,huang2007} and requires keeping at least the $m=1$ mode.\n\n\\subsection{Plasma wakefield excitation}\n\nWe start by simulating linear wakefield excitation. The linear regime refers to the case that the peak density of the drive beam $n_b$ is much smaller than the background plasma density $n_p$, so that the drive beam only introduces a weak perturbation to the plasma and the background electrons oscillate in a nearly sinusoidal fashion. In this case, the drive beam has a bi-Gaussian density profile with a spot size $k_p\\sigma_r=2.0$, bunch length $k_p\\sigma_z=0.5$, and peak density $n_b\/n_p=0.1$, where $k_p^{-1}$ is the plasma skin depth where $n_b=\\frac{N}{(2\\pi)^{3\/2}} \\exp[-(\\frac{r^2}{2\\sigma^2_r}+\\frac{z^2}{2\\sigma^2_z})]$ and $N$ is the number of particles in the bunch. Since this scenario possesses azimuthal symmetry, we only include the $m=0$ mode in QPAD which is equivalent to a 2D r-z simulation using codes such as WAKE or LCODE. In the QuickPIC and OSIRIS simulations, the cell size is $\\Delta_x=\\Delta_y=0.0234\\ k_p^{-1}, \\Delta_z=0.0195\\ k_p^{-1}$. In the QPAD simulation, $\\Delta_r=0.0234\\ k_p^{-1}, \\Delta_z=0.0195\\ k_p^{-1}$. The drive beams are initialized with $128\\times128\\times256$ particles in $x, y$ and $z$ for the QuickPIC simulation and with $128\\times32\\times256$ particles in $r,\\phi$ and $z$ for the QPAD simulation. For the plasma, we use $2\\times2$ particles per 2D cell in QuickPIC and uniformly distribute $2\\times32$ particles within a ring of width $\\Delta_r$ in QPAD. In OSIRIS, $2\\times2\\times2$ particles per 3D cell are used to initialize both the plasma and beam.\n\nThe simulation results are shown in Fig. \\ref{fig:linear_wake}. In figure \\ref{fig:linear_wake}(a) and (b), we compare the plasma electron density and $E_z$ field between QuickPIC and QPAD runs. The drive beams, whose centers reside at $\\xi=2$, move downward and are not displayed in these figures. Figure \\ref{fig:linear_wake}(c) compares the lineouts of $E_z$ on the $r=0$  axis between QPAD, QuickPIC and OSIRIS. Here, only one predictor-corrector iteration is conducted in QPAD and this already gives excellent agreement with QuickPIC and OSIRIS. We also conducted convergence tests for the predictor-corrector loop by iterating 1, 3 and 5 times. We found in this scenario, the predictor-corrector loop converges so rapidly that only one iteration is sufficient to reach the desired simulation accuracy.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{linear_wake.pdf}\n\\caption{Comparison of beam-driven wakefield in linear regime between OSIRIS, QuickPIC and QPAD. (a) Background electron density. (b) $E_z$ field. (c) On-axis lineouts of $E_z$ fields from OSIRIS, QuickPIC and QPAD.}\n\\label{fig:linear_wake}\n\\end{figure}\n\nNext, we simulate drive beam parameters for which a nonlinear plasma wakefield is excited. In this case the peak density of the beam is much larger than the plasma density, \\emph{i.e.}, $n_b\\gg n_p$.\nHere, we show an example for which $n_b\/n_p=4$, $k_p\\sigma_r=0.25$, $\\Lambda\\equiv(n_b\/n_p)(k_p\\sigma_r)^2=0.25$ and keep other numerical parameters the same as those in the linear regime case. In the nonlinear regime, the $E_z$ on axis now looks similar to a sawtooth wave as shown in fig. \\ref{fig:nonlinear_wake}(c). In the region where the background plasma electrons are fully evacuated by the drive beam (from $\\xi=3$ to $\\xi=7$), the $E_z$ field almost drops linearly to its minimum at the rear of the first ion bubble. From fig. \\ref{fig:nonlinear_wake}, we can see that QPAD with only one predictor-corrector iteration still gives results in almost perfect agreement with OSIRIS and QuickPIC. Similarly to the convergence test for the linear regime, the predictor-corrector iteration is found to converge rapidly. Running the iteration more than once does not make an observable difference to the simulation results.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{nonlinear_wake.pdf}\n\\caption{Comparison of beam-driven wakefield in nonlinear regime between OSIRIS, QuickPIC and QPAD. (a) Background electron density. (b) $E_z$ field. (c) On-axis lineouts of $E_z$ fields from OSIRIS, QuickPIC and QPAD.}\n\\label{fig:nonlinear_wake}\n\\end{figure}\n\nBesides an electron beam, a very short positron or proton beam can also excite a bubble-like plasma wake. Due to the attractive force from the positron bunch, the background electrons are ``sucked in'' first by the drive beam rather than ``blown out'' as is the case for an electron beam driver. This leads to the background electrons forming a density peak at the front of the first bucket, and the $E_z$ field being negative in that region. After the plasma electrons collapse to the axis, they then overshoot and eventually form a blowout type wake in the second wavelength. In figure \\ref{fig:positron}(a) and (b), a bi-Gaussian positron beam with $n_b\/n_p=2.5,\\ k_p\\sigma_r=0.8,\\ k_p\\sigma_\\xi=0.46$ and the center resides at $\\xi=3$ moves downward. Again, we use only one predictor-corrector iteration to achieve good agreements with the results of QuickPIC and OSIRIS.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{positron.pdf}\n\\caption{Comparison of positron-beam-driven wakefield between OSIRIS, QuickPIC and QPAD. (a) Background electron density. (b) $E_z$ field. (c) On-axis lineouts of $E_z$ fields from OSIRIS, QuickPIC and QPAD.}\n\\label{fig:positron}\n\\end{figure}\n\n\\subsection{Hosing instability}\n\nIn this section, we present a simulation of what is called the hosing instability in PWFA \\cite{huang2007}. The hosing instability is one of the major impediments for PWFA and can lead to beam breakup. Although an azimuthally symmetric r-z code such as WAKE and LCODE is very efficient to model PWFA, it cannot be used to investigate the physics involving asymmetries such as the hosing instability. For hosing we only compare QPAD against QuickPIC. The drive beam has a bi-Gaussian profile with a peak density $n_b\/n_p=93.5$, an rms spot-size $k_p\\sigma_r=0.14$ and an rms bunch length $k_p\\sigma_z=0.48$ which corresponds to $\\Lambda\\simeq1.8$. The trailing beam parameters are $n_b\/n_p=56$, $k_p\\sigma_r=0.14$ and $k_p\\sigma_z=0.24$.  For both the plasma and the beams there are 16 macro-particles distributed in $\\phi$ while for the plasma there are 2 macro-particles per r-z cell. Within the region $[-5\\sigma_r,+5\\sigma_r]\\times[-5\\sigma_z,+5\\sigma_z]$ the drive beam and trailing beam have $128\\times512$ and $128\\times256$ particles respectively, and have 16 particles azimuthally. The drive beam is initialized axisymmetrically while the trailing beam has a small centroid offset of $0.038\\ k_p^{-1}$ in $x$-direction. For the full 3D QuickPIC simulation, the plasma has $2\\times2\\times2$ particles per cell and the drive beam and trailing beam have $128\\times128\\times512$ and $128\\times128\\times256$ particles within the $5\\sigma$ rectangular block. The initial longitudinal proper velocity corresponds to $\\gamma \\beta_z=20000$ for both the drive and trailing beams.  In the QPAD simulation, modes $m=0, 1, 2$, and 3 are included.  Figure \\ref{fig:hosing_density} shows the density distribution with the background plasma electrons and beams colored blue and red respectively. The snapshots were taken at $\\omega_p t=20000$. It can be seen that there is excellent agreement between QPAD and QuickPIC for the motion of the trailing beam even for this nonlinear problem.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{hosing_density.pdf}\n\\caption{Density distribution of plasma electrons and beams in (a) full 3D QuickPIC and (b) QPAD simulations.}\n\\label{fig:hosing_density}\n\\end{figure}\n\nA more careful comparison between the hosing results is obtained by investigating the beam centroid oscillation during the entire acceleration distance for different beam slices. Figure \\ref{fig:hosing_centroid}(a)-(c) plots the centroid oscillation for three slices, residing at $+\\sigma_z,\\ 0$ and $-\\sigma_z$ with respect to the beam center $\\xi_0$. The centroid is defined as $\\frac{1}{N}\\sum x_i$ where the sum is taken over all particles within a slice at $z \\pm 0.1k_p^{-1}$ and $N$ is the number of particles. The amplitude of the centroid oscillation for the slice closer to the beam head [figure \\ref{fig:hosing_centroid}(c)] remains nearly constant in $s$, the amplitude grows in $s$ with a larger growth rate the farther the slice is behind the center of the beam [figure \\ref{fig:hosing_centroid}(a) and (b)]. This qualitatively agrees well with the theoretical prediction on the instability growth. Except for a slight phase difference that is evident for larger values of $s$, there is excellent agreement between QPAD and full 3D QuickPIC simulations. These differences may be due to the truncation of the azimuthal mode expansion at $m=3$. We emphasize that a code such as QPAD is also a powerful too for carrying out large parameter scans even if the results are not quantitatively correct.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{hosing_centroid.pdf}\n\\caption{Beam centroid oscillation of slice residing at (a) $+\\sigma_z$, (b) 0 and (c) $-\\sigma_z$ with respect to the beam center $\\xi_0$.}\n\\label{fig:hosing_centroid}\n\\end{figure}\n\n\\section{Algorithm complexity}\n\\label{sec:performance}\n\n\nThe azimuthal-decomposition-based algorithm has the potential to greatly reduce the computational requirements without much loss in accuracy when modeling 3D physics when the problem only has low order azimuthal asymmetry. This is because it requires fewer grid points and hence few particles. We can make a straightforward estimation of the speedup over a full 3D quasi-static code.\n\nIn QuickPIC, the fields are solved on a 2D slab (usually a square) with $n_\\text{mesh}=N^2$ grid points, so the cost of the Poisson solver is $O(N^2\\log(N))$ assuming the fast FFT method is used. In QPAD, we solve fields on a 1D mesh with $n_\\text{mesh}=N\/2$ grid points for $2m_\\text{max}+1$ components ($m=0$ mode and real\/imaginary parts for $m>0$ modes) where $m_\\text{max}$ is the index of the highest azimuthal mode that is kept.  Therefore, the cost of the Poisson solver is $(2m_\\text{max}+1)O[(N\/2)\\log(N\/2)]$ using the cyclic reduction method. The speedup for the field solve will therefore scale as $\\sim O(N)\/(m_\\text{max}+\\frac{1}{2})$ compared with the FFT method used in QuickPIC. In QuickPIC, a total number of $N^2N_{\\text{ppc},x}N_{\\text{ppc},y}$ macro-particles for plasma species are used where $N_{\\text{ppc},\\eta},\\ (\\eta=x,y)$ denotes the particle number per cell in the $\\eta$-direction. In QPAD, there are only $NN_{\\text{ppc},r}N_{p,\\phi}\/2$ macro-particles for each plasma species, where $N_{\\text{ppc},r}$ is number of particles per $r$-$z$ cell  and $N_{p,\\phi}$ is the number of particles distributed over $0<\\phi<2\\pi$. Assuming the computational cost of pushing particles is proportional to the total macro-particle number, the speedup therefore scales as $2NN_{\\text{ppc},x}N_{\\text{ppc},y}\/(N_{\\text{ppc},r}N_{p,\\phi})\\sim O(N)$. For a majority of PWFA problems, the configuration with $m_\\text{max}\\leq2$ and particle number $N_{p,\\phi}\\sim10,\\ N_{\\text{ppc},r}\\sim N_{\\text{ppc},x}$ or $N_{\\text{ppc},y}$ are enough to capture the dominant azimuthal asymmetry to effectively simulate the physics with nearly round drive beams, so that considerable speedup can be achieved for typical numerical parameters. The goal of this paper is to describe how to implement an azimuthal mode expansion into a quasi-static PIC code. Issues with respect to optimization will be addressed in future publications. The parallelization in QPAD is also similar to that in QuickPIC. The code is parallelized using MPI to run on distributed memory clusters, which is implemented by means of spatial decomposition in r and z dimensions. However, owing to the basic numerical scheme of a quasi-static code, the parallelization in $r$ direction differs essentially from the that in $\\xi$ direction. The parallelization in $r$ is similar to that in full explicit PIC codes with the macro-particles exchange between neighboring processors. The interprocess exchange of field values at the domain boundaries is handled by the built-in routines of Hypre library. In the $\\xi$ direction, we use pipelining algorithm to allow the transverse process slabs to run asynchronously, which can significantly inhibit the idle time.\n\n\n\n\\section{Conclusion}\n\nWe have describe QPAD, a new quasi-static PIC code that uses the azimuthal Fourier decomposition for the fields. The new code utilizes the workflow and routines of QuickPIC in which a 2D code for evolving the plasma particles in a time like variable $\\xi$ is embedded into a 3D code that advances beam particles in a time like variable $s$. In QPAD, all the field components are decomposed into a few Fourier harmonics in $\\phi$. In the 2D part of the code each amplitude depends on $r$ and evolves in $\\xi$. Therefore, in this part of the code the fields are only defined on a 1D grid in $r$. The quasi-static version of  Maxwell's equations for each harmonic amplitude are therefore one-dimensional, making the new code much faster. A full set of Poisson-like equations that exactly correspond to those used in the full 3D QuickPIC are written in cylindrical geometry. A full set of 1D Poisson equations in $r$ are solved for the Fourier amplitudes in $\\phi$ for the relevant fields. To simplify the calculation, we introduced linear combinations of the complex amplitudes, $B^m_\\pm$, to decouple the equations for $B^m_r$ and $B^m_\\phi$. Open (free) boundary conditions are implemented for all the fields. For the particle module, the macro-particles are distributed and advanced in $\\xi$ in a 2D space $(r,\\phi)$.  A predictor-corrector routine is described. A novel deposition method for $\\PP{\\bm{J}}{\\xi}$, $\\bm{J}$ and $\\rho-J_z$ for each harmonics is described and implemented. This scheme does not require updating the particle positions to obtain $\\PP{\\bm{J}}{\\xi}$ which reduces the complexity of the predictor corrector routine.  The new code was benchmarked and compared against results from 3D OSIRIS and QuickPIC for a few sample cases. Excellent agreement was found for both wake excitation of plasma wave wakes from particle beam drivers (electrons and positrons) and for the electron hosing instability. Directions for future work include optimizing the field solver to reduce the across-node data communication, adding multi-threading features (OpenMP), and implementing more physics including field-ionization, radiation reaction, and the ponderomotive guiding center model for a laser.\n\n\\section*{Acknowledgments}\n\n\nWork supported by the U.S. Department of Energy under SciDAC FNAL subcontract 644405, DE-SC0010064, and contract number DE-AC02-76SF00515, and by the U.S. National Science Foundation under NSF 1806046. The simulations were performed on the UCLA Hoffman 2 and Dawson 2 Clusters, the resources of the National Energy Research Scientific Computing Center, and the Super Computing Center of Beijing Normal University. \n\n\\begin{appendix}\n\\section{Implementation of free boundary conditions for electromagnetic fields}\n\\label{sec:app:free_bnd}\n\nIn this appendix, we describe the implementation of the open (free) boundary conditions used to solve Eqs. (\\ref{eq:pois_psi_m})-(\\ref{eq:pois_bz_m}) and (\\ref{eq:pois_ez_m}). Figure \\ref{fig:boundary} shows the grid setup for solving the fields with total $N$ grid points within the solution region. The dashed line defines the boundary and the the physical domain. It is assume that outside of this region there is vacuum out to infinity.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{boundary.pdf}\n\\caption{Grid points layout in $r$-direction.}\n\\label{fig:boundary}\n\\end{figure}\n\nThe basic idea is to obtain the analytic solution in vacuum by solving Laplace equations and then applying solutions at the boundary. We first consider the scalar Laplace equation,\n\\begin{equation}\n\\label{eq:laplace_scalar}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m U^m=0,\\quad\\text{for}\\ r>R,\n\\end{equation}\nwhere $U^m$ represents the $m$th mode of $\\psi$, $A_z$, $B_z$ and $E_z$. It has the solution\n\\begin{equation}\n\\label{eq:laplace_sol_0}\nU^0=C_{U,0}+D_{U,0}\\ln(r)\n\\end{equation}\nand\n\\begin{equation}\nU^{m>0}=C_{U,m} r^{-m}+D_{U,m} r^m.\n\\end{equation}\nThe determination of the constants $C_{U,m}$ and $D_{U,m}$ differs depending on the types of fields. For $\\psi$, it can be shown that $D_{\\psi,0}=0$. By applying Gauss's theorem to Eq. (\\ref{eq:poisson_psi}) and considering a circular  region $S$ of integration with a radius greater than $R$, leads to\n\\begin{equation}\n\\label{eq:gauss_theorem}\n\\oint_{\\partial S} \\nabla_\\perp\\psi\\ dl=2\\pi r\\PP{\\psi^0}{r}=-\\int_S (\\rho-J_z) dS.\n\\end{equation}\nNote that the $m>0$ modes of $\\psi$ do not contribute to the integral on the left because of the presence of the term $e^{\\mathrm{i} m\\phi}$. From the continuity equation under the QSA\n\\begin{equation}\n\\PP{}{\\xi}(\\rho-J_z)+\\nabla_\\perp\\cdot\\bm{J}_\\perp=0.\n\\end{equation}\nand using the fact that $\\bm{J}_\\perp$ vanishes at the boundary of the surface integral $\\partial S$, we have\n\\begin{equation}\n\\PP{}{\\xi}\\int_S (\\rho-J_z)dS=0.\n\\end{equation}\nwhich indicates this integral is zero for any $\\xi$ because it is initially zero (neutral plasma). Therefore, according to Eq. (\\ref{eq:gauss_theorem}), we have $2\\pi r\\PP{\\psi^0}{r}|_{r>R}=0$ which gives $D_{\\psi,0}=0$ by inserting Eq. (\\ref{eq:laplace_sol_0}). Requiring $\\psi\\rightarrow 0$ while $r\\rightarrow 0$, we can determine that $C_{\\psi,0}=0$ and $D_{\\psi,m}=0\\ (m>0)$, and thus the solution in the vacuum has the form\n\\begin{equation}\n\\psi^0=0,\\quad\n\\psi^m=\\frac{C_{\\psi,m}}{r^m}.\n\\end{equation}\n\nFor the longitudinal component of beam's vector potential $A_z$, $D_{A_z,0}\\neq0$ because the charge of the beam is apparently non-neutralized. Applying the natural boundary condition $B_{\\phi,\\text{beam}}=\\PP{A_z}{r}\\rightarrow0$ when $r\\rightarrow0$ and ignoring the arbitrary constant, we have\n\\begin{equation}\nA_z^0=D_{A_z,0}\\ln(r),\\quad\nA_z^m=\\frac{C_{A_z,m}}{r^m}.\n\\end{equation}\n\nFor $B_z$ and $E_z$, the only constraint is $B_z,E_z\\rightarrow 0$ when $r\\rightarrow 0$, so that\n\\begin{equation}\nE_z^0=0,\\quad\nE_z^m=\\frac{C_{E_z,m}}{r^m}\n\\end{equation}\nand\n\\begin{equation}\nB_z^0=0,\\quad\nB_z^m=\\frac{C_{B_z,m}}{r^m}.\n\\end{equation}\n\nThe transverse magnetic fields induced by the plasma satisfy the coupled Laplace equations in the vacuum,\n\\begin{align}\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_r^m-\\frac{B_r^m}{r^2}-\\frac{2\\mathrm{i} m}{r^2}B_\\phi^m&=0, \\\\\n\\mathop{}\\!\\mathbin\\bigtriangleup_m B_\\phi^m-\\frac{B_\\phi^m}{r^2}+\\frac{2\\mathrm{i} m}{r^2}B_r^m&=0.\n\\end{align}\nIt can be verified that the general solution can be written as\n\\begin{equation}\nB_r^0=\\frac{C_{B_r,0}}{r},\\ B_r^m=\\frac{C_{B_r,m}}{r^{m+1}}+D_{B_r,m}r^{m-1},\n\\end{equation}\nand\n\\begin{equation}\nB_\\phi^0=\\frac{C_{B_\\phi,0}}{r},\\ B_\\phi^m=\\frac{C_{B_\\phi,m}}{r^{m+1}}+D_{B_\\phi,m}r^{m-1}.\n\\end{equation}\nHere, $D_{B_r,m}=D_{B_\\phi,m}=0$ because of the natural boundary conditions that $B_r,B_\\phi\\rightarrow 0$ when $r\\rightarrow 0$.\n\nAfter obtaining the analytical solution for each field (components) in the vacuum, we derive the finite difference form of the boundary conditions used for solving the discrete Poisson-like equations. For an arbitrary field $U^m$, the value on the ghost cell can be evaluated through Taylor expansion (central difference)\n\\begin{equation}\nU^m_{N+1}=U^m_N+\\left.\\PP{U^m}{r}\\right|_N\\Delta_r+O(\\Delta_r^2).\n\\end{equation}\nThe derivative at $r_{N+\\frac{1}{2}}$ (note that $r_{N+\\frac{1}{2}}=R$) is evaluated using the analytical formula. For $A_z^0$,\n\\begin{equation}\nA^0_{z,N+1}\\simeq A^0_{z,N}+\\frac{D_{A_z,0}}{R}\\Delta_r\\simeq A_{z,N}^0+\\frac{\\Delta_r}{R\\ln R}A^0_{z,N+1}\n\\end{equation}\ntherefore\n\\begin{equation}\nA^0_{z,N+1}\\simeq\\left(1+\\frac{\\Delta_r}{R\\ln(R)}\\right)A^0_{z,N}.\n\\end{equation}\nSimilarly, for $m>0$ modes of $\\psi,A_z,B_z$ and $E_z$, we can obtain\n\\begin{equation}\n\\begin{pmatrix}\n\\psi \\\\ A_z \\\\ B_z \\\\ E_z\n\\end{pmatrix}^m_{N+1}\\simeq\\left(1-\\frac{m\\Delta_r}{R}\\right)\n\\begin{pmatrix}\n\\psi \\\\ A_z \\\\ B_z \\\\ E_z\n\\end{pmatrix}^m_N\n\\end{equation}\nand for all the modes of $B_r$ and $B_\\phi$ associated with plasma\n\\begin{equation}\n\\begin{pmatrix}\nB_r \\\\ B_\\phi\n\\end{pmatrix}^m_{N+1}\\simeq\\left(1-\\frac{(m+1)\\Delta_r}{R}\\right)\n\\begin{pmatrix}\nB_r \\\\ B_\\phi\n\\end{pmatrix}^m_N.\n\\end{equation}\nAs $B_\\pm^m$ rather that $B_r^m$ and $B_\\phi^m$ are directly solved in QPAD, we need to perform the linear transformation  $B_\\pm^m=B_r^m\\pm\\mathrm{i} B_\\phi^m$ on both sides of the above equation to obtain  the boundary condition for $B_\\pm^m$\n\\begin{equation}\nB_{\\pm,N}^m\\simeq\\left(1-\\frac{(m+1)\\Delta_r}{R}\\right)B_{\\pm,N+1}^m.\n\\end{equation}\n\n\\end{appendix}\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{}}\n\n\n\\newcommand{\\subsection}{\\subsection}\n\n\n\n\\newcommand{\\bC^\\times}{{\\mathbb C}^\\times}\n\n\n\n\\newcommand{\\operatorname{Ker}}{\\operatorname{Ker}}\n\n\\newcommand{\\operatorname{Exp}}{\\operatorname{Exp}}\n\n\n\n\n\n\n\n\n\n\n\\title[Centralizers of generic elements of Newton strata]{Centralizers\nof generic elements of Newton strata in the adjoint quotients of reductive\ngroups}\n\\author[Mitya Boyarchenko and Maria Sabitova]{Mitya\nBoyarchenko\\address{Mitya Boyarchenko: University of Chicago, \\hfill\\newline\nDepartment of Mathematics, Chicago, IL 60637} and Maria Sabitova\\address{Maria\nSabitova: University of Illinois at Urbana-Champaign, \\hfill\\newline Department\nof Mathematics, Urbana, IL 61801}}\n\n\\thanks{The research of M.B. was partially supported by NSF grant\nDMS-0401164.  \\hfill\\newline {\\em Email addresses:}\\ \\ {\\tt\nmitya@math.uchicago.edu} (M.B.), \\ \\ {\\tt sabitova@math.upenn.edu} (M.S.)}\n\n\n\n\n\n\n\n\\begin{document}\n\n\n\n\n\n\n\n\\begin{abstract}\nWe study the Newton stratification of the adjoint quotient of a connected split\nreductive group $G$ with simply connected derived group over the field $F={\\mathbb C}((\\epsilon))$ of formal Laurent series. Our main result describes the\ncentralizer of a regular semisimple element in $G(F)$ whose image in the\nadjoint quotient lies in a certain generic subset of a given Newton stratum.\nOther noteworthy results include analogues of some results of Springer on\nregular elements of finite reflection groups, as well as a geometric\nconstruction of a well known homomorphism $\\psi:P(R^\\vee)\/Q(R^\\vee)\\to W(R)$ defined\nfor every reduced and irreducible root system $R$.\n\\end{abstract}\n\n\\maketitle\n\n\\setcounter{tocdepth}{1}\n\n\\tableofcontents\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Introduction}\n\nIf $F={\\mathbb C}((\\epsilon))$ is the field of formal Laurent series and $G=GL_n(\\overline{F})$, it\nis well known that a semisimple element of $G$ is conjugate to an element lying\nin $GL_n(F)$ if and only if the coefficients of its characteristic polynomial\nlie in $F$, and, conversely, every monic polynomial of degree $n$ over $F$\nwhose constant term in nonzero is the characteristic polynomial of a semisimple\nelement of $G$ lying in $GL_n(F)$. Hence the $F$-points of the adjoint quotient\n${\\mathbb A}$ of $G$ are identified with the set  of such\npolynomials: ${\\mathbb A}(F)\\cong F^{n-1}\\times F^\\times$. We can partition ${\\mathbb A}(F)$ into a disjoint union of subsets\n${\\mathbb A}(F)_{\\nu}$ consisting of polynomials of ${\\mathbb A}(F)$ having the same Newton\npolygon $\\nu$, which gives the {\\em Newton stratification} of ${\\mathbb A}(F)$.\n\n\\medbreak\n\nIn his paper \\cite{k} R.~Kottwitz generalizes the notion of Newton\nstratification to a general connected split reductive group $G$ over $F$ whose\nderived group is simply connected. One of the main goals of this paper is to\nanswer a question (also formulated by Kottwitz) which arises naturally in this\ncontext. Namely, we prove that for every Newton stratum\n${\\mathbb A}(F)_\\nu\\subset{\\mathbb A}(F)$ and for every regular semisimple element $\\gamma\\in\nG(F)$ whose image in ${\\mathbb A}(F)$ lies within a certain open dense subset of\n${\\mathbb A}(F)_\\nu$, the centralizer $G_\\gamma$ is isomorphic to the twist of a split\nmaximal torus $A\\subset G$ by an element $w=w(\\nu)$ of the Weyl group of $G$\nwith respect to $A$. The element $w$ is independent of the choice of $\\gamma$ and admits a\nrather simple description in terms of $\\nu$ (see Theorem \\ref{t:1}). The\ndefinition of the Newton stratification appears in Section \\ref{s:notation}\ntogether with some other preliminaries, while the statement and the proof of\nour main result occupy Sections \\ref{s:main} and \\ref{s:systems}.\n\n\\medbreak\n\nIn the course of the proof we encountered several questions about semisimple\ngroups over ${\\mathbb C}$ that, to the best of our knowledge, have not been studied\nbefore. In particular, if $G$ is a simple, connected and simply connected\nalgebraic group over ${\\mathbb C}$ with maximal torus $A\\subset G$, we establish in\nSection \\ref{s:Springer} natural analogues of some of the results of\nT.~Springer \\cite{s} in the nonlinear context of the Weyl group $W=W(G,A)$\nacting on $A$. More precisely, we prove that, given an element $x$ of the center $Z(G)$ of $G$,\nthere exist $g\\in W$ and a regular element $u\\in A$ satisfying $g(u)=xu$; in\naddition, this condition determines the element $g$ uniquely up to conjugation,\nand the eigenvalues of the action of $g$ on the Lie algebra $\\operatorname{Lie}(A)$ can be\ncomputed easily in terms of $x$. Aside from the applications appearing in this\npaper, the usefulness of these results is demonstrated by Remark\n\\ref{r:kottwitz}.\n\n\\medbreak\n\nSection \\ref{s:geometric} is devoted to the study of a certain important\nhomomorphism $\\psi:Z(G)\\to W$ (which already enters the formulation of our\nfirst main result). This homomorphism is well defined up to conjugation by\nelements of $W$. So far the only known (to us) description of this homomorphism\nwas purely algebraic. Our last main result is Theorem \\ref{t:monodromy}, which\nprovides a geometric (or, if the reader wishes, topological) description of\n$\\psi$ as the monodromy of a certain $W$-torsor over a manifold whose\nfundamental group is identified with $Z(G)$, constructed by restricting the\nSpringer covering $\\widetilde{G}^{rs}\\to G^{rs}$ to the set of regular elements\nof a maximal compact subgroup of $G$.\n\n\n\n\n\\subsection*{Acknowledgements} We would like to express our deepest\ngratitude to Robert Kottwitz for introducing us to this subject and suggesting\nthe statement of one of our main results, as well as carefully reading our article. We would also like to thank him for\nsharing the preprint \\cite{k} before it was made available to the general\naudience, for his time, for numerous stimulating discussions, and in particular\nfor pointing out a way of greatly simplifying the proof of the first part of\nTheorem \\ref{t:monodromy}. We are grateful to Misha Finkelberg for useful and\ninspiring conversations, for referring us to the paper \\cite{ss}, and for\nexplaining to us Zakharevich's construction that appears in\n\\S\\ref{ss:monodromy}. We are indebted to Jim Humphreys, David Kazhdan, George\nLusztig, Jean-Pierre Serre, and Tonny Springer for very helpful e-mail\ncorrespondence. Part of the research was conducted when the second author was a\npostdoc at MSRI in the Spring of 2006, and she would like to thank the\nInstitute for its hospitality.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{General facts and notation}\\label{s:notation}\n\n\nSince the main reference for the first part of this paper is \\cite{k}, we will\nkeep the same notation as in that article. In this section we review this\nnotation and recall some results that will be used in the remainder of the\npaper. The reader familiar with \\cite{k} may wish to skip directly to Section\n\\ref{s:main} and refer back to this one as the need arises.\n\n\n\n\\subsection{The setup}\\label{ss:setup} Let $G$ be a connected reductive group\nover ${\\mathbb C}$ of rank $n$. We assume that the derived group $G_{sc}=[G,G]$ of $G$\nis simply connected, and let $l$ be its rank (so that $l\\leq n$). The principal\nexample to keep in mind is $G=GL_n$. We fix a maximal torus $A\\subset G$ and a\nBorel subgroup $B\\subset G$ containing $A$, and write $A_{sc}=A\\cap G_{sc}$,\nwhich is a maximal torus of $G_{sc}$. We let $R\\subset X^*(A)$ denote the root\nsystem of $G$ with respect to $A$, and $\\Delta=\\{\\alpha_i\\}_{i\\in I}\\subset R$ the\nbasis of simple roots determined by $B$, where $I=\\{1,2,\\dotsc,l\\}$. Further,\nwe let $Q(R)$ and $Q(R^\\vee)$ denote the root lattice and the coroot lattice of\n$R$, and we let $P(R)$ and $P(R^\\vee)$ denote the dual lattices to $Q(R^\\vee)$\nand $Q(R)$, respectively. It is known that $P(R)$ can be canonically identified\nwith $X^*(A_{sc})$, and $Q(R^\\vee)$ can be canonically identified with\n$X_*(A_{sc})$. We will view these identifications as equalities. Finally, let\n$\\{\\varpi_i\\}_{i\\in I}$ denote the basis of fundamental weights of\n$P(R)=X^*(A_{sc})$ which is dual to the basis of $Q(R^\\vee)$ consisting of the\nsimple coroots $\\{\\alpha_i^\\vee\\}_{i\\in I}$, and let $\\omega_1,\\ldots,\\omega_n\\in\nX^*(A)$ be a ${\\mathbb Z}$-basis chosen as in \\cite{k} (\\S 1.3, p.~3). Namely,\n$\\omega_1,\\ldots,\\omega_l$ are preimages of the fundamental weights\n$\\varpi_1,\\ldots,\\varpi_l\\in X^*(A_{sc})$ under the natural map\n$\\operatorname{Res}:X^*(A)\\twoheadrightarrow X^*(A_{sc})$ and $\\omega_{l+1},\\ldots,\\omega_n$ is a\n${\\mathbb Z}$-basis of the kernel of $\\operatorname{Res}$.\n\n\n\n\n\n\\subsection{The adjoint quotient of $G$}\\label{ss:adj-quotient}\nLet $W=W(G,A)=N_G(A)\/A$ be the Weyl group of $G$ with respect to $A$. For each\n$\\lambda\\in X^*(A)$ we write $e^{\\lambda}$ for $\\lambda$ viewed as an element of the group\nalgebra ${\\mathbb C}[X^*(A)]$ of $X^*(A)$ over ${\\mathbb C}$; note that\n${\\mathbb C}[X^*(A)]\\cong{\\mathbb C}[A]$, the coordinate ring of $A$. Put\n$$\nc_i:=\\sum_{\\lambda\\in W\\omega_i}e^{\\lambda}\\in {\\mathbb C}[X^*(A)]^W,\\quad i\\in\\{1,\\ldots,n \\},\n$$\nwhere $W\\omega_i$ denotes the $W$-orbit of $\\omega_i$ in $X^*(A)$. Let\n${\\mathbb A}:={\\mathbb G}_a^l\\times{\\mathbb G}_m^{n-l}$. It is well-known that the map $c:A\\longrightarrow{\\mathbb A}$ defined\nby $$a\\mapsto (c_1(a),\\ldots,c_n(a))$$ induces an isomorphism of algebraic\nvarieties between $A\/W$ and ${\\mathbb A}$ (see \\cite{k}, \\S1.5, p. 5, for example). We will often\nidentify $A\/W$ with ${\\mathbb A}$ via $c$. Note that, in particular, we obtain an\nidentification $A(\\overline{F})\/W\\cong{\\mathbb A}(\\overline{F})$ for any algebraically\nclosed field $\\overline{F}$ containing ${\\mathbb C}$.\n\n\\medbreak\n\nThe variety ${\\mathbb A}$ is called the {\\em adjoint quotient} of $G$, because if we\nlet $G$ act on itself by conjugation, Chevalley's restriction theorem implies\nthat the restriction map ${\\mathbb C}[G]\\to{\\mathbb C}[A]$ induces an isomorphism of algebras\n${\\mathbb C}[G]^G\\cong{\\mathbb C}[A]^W$, and hence we obtain an isomorphism of algebraic\nvarieties $A\/W\\cong G\/\\!\/(\\operatorname{Ad} G)$.\n\n\n\n\n\n\n\n\n\\subsection{The base field $F$}\\label{ss:base-field}\nLet $F={\\mathbb C}((\\epsilon))$ be the field of formal Laurent power series\nover ${\\mathbb C}$ in an indeterminate $\\epsilon$.  For each $n\\in{\\mathbb N}$ we let $\\epsilon^{1\/n}\\in\\overline{F}^{\\times}$\nbe a fixed $n$-th root of $\\epsilon$\nsuch that\n$$\n\\left(\\epsilon^{\\frac{1}{mn}}\\right)^{m}=\\epsilon^{1\/n},\\quad\n\\forall m,n\\in{\\mathbb N}.\n$$\nIt is known that $\\overline{F}=\\bigcup_{n\\in{\\mathbb N}}{\\mathbb C}((\\epsilon^{1\/n}))$. Thus, the element\n$\\sigma\\in\\Gamma=\\operatorname{Gal}(\\overline{F}\/F)$ given by\n$$\n\\sigma(\\epsilon^{1\/n})=\\exp\\left(\\frac{2\\pi\n\\sqrt{-1}}{n}\\right)\\epsilon^{1\/n},\\quad\\forall n\\in{\\mathbb N},\n$$\nis a topological generator of $\\Gamma$, i.e., it determines an\nisomorphism $\\widehat{{\\mathbb Z}}\\rarab{\\simeq}\\Gamma$. We also fix the valuation $\\operatorname{val}$ on $\\overline{F}$ such that\n$$\n\\operatorname{val}(\\epsilon^{1\/n})=-1\/n,\\quad\\forall n\\in{\\mathbb N}, \\quad\\text{ and }\\,\\,\\operatorname{val}(0)=-\\infty.\n$$\n\n\\medbreak\n\nWe may view $G$ and $A$ as algebraic groups over $F$ by extending scalars. This\nshould cause no confusion, since in the remainder of this section, as well as\nin Sections \\ref{s:main} and \\ref{s:systems} we always think of $G$ and $A$ as\nalgebraic groups over $F$, whereas in Sections \\ref{s:Springer} and\n\\ref{s:geometric} we will only work with algebraic groups over ${\\mathbb C}$. We will\nconsider ${\\mathbb A}(F)=F^l\\times (F^\\times)^{n-l}$ as a topological space with\nrespect to the topology induced from $F^n$, where $F$ is identified with a\nproduct of countably many copies of ${\\mathbb C}$ and the closed sets in $F^n$ are\ndefined by polynomial equations over ${\\mathbb C}$ in finitely many coordinates.\n\n\n\n\n\\subsection{The map $f:G^{rs}(F)\\longrightarrow{\\mathbb A}(F)$}\\label{subsec:f}\nLet $G^{rs}(F)$ denote the set of regular semisimple elements of $G$ defined\nover $F$. We will need the map $f:G^{rs}(F)\\longrightarrow{\\mathbb A}(F)$ defined in the following\nway. Let $\\gamma\\in G^{rs}(F)$ be a regular semisimple element of $G$, i.e.,\n$\\gamma\\in G(F)$ and the centralizer $G_{\\gamma}$ of $\\gamma$ in $G$ is a maximal torus\n(defined over $F$). There is $g\\in G(\\overline{F})$ which conjugates $\\gamma$ to an element\n$\\gamma_0\\in A(\\overline{F})$. Note that $\\gamma_0$ is only determined up to the $W$-action.\nNamely, if $g_1\\gamma g_1^{-1}\\in A(\\overline{F})$ and $g_2\\gamma g_2^{-1}\\in A(\\overline{F})$, then\nthe element $g_1g_2^{-1}\\in G(\\overline{F})$ conjugates $g_2\\gamma g_2^{-1}$ into $g_1\\gamma\ng_1^{-1}$. Since $\\gamma$ is regular and semisimple,  we have\n$$\nA(\\overline{F})=G_{g_1\\gamma g_1^{-1}}(\\overline{F})=G_{g_2\\gamma g_2^{-1}}(\\overline{F}),\n$$\nhence $g_1g_2^{-1}\\in N_G(A)(\\overline{F})$, where $N_G(A)$ denotes the normalizer of\n$A$ in $G$. Thus, $g_1\\gamma g_1^{-1}$ and $g_2\\gamma g_2^{-1}$ are in the same\n$W$-orbit. So we get a well-defined map $f:G^{rs}(F)\\longrightarrow{\\mathbb A}(\\overline{F})$ given by\n$$\nf(\\gamma)=c(\\gamma_0).\n$$\nWe claim that, in fact, $f(\\gamma)\\in{\\mathbb A} (F)$. To check the claim we need to show\nthat the $W$-orbit of $\\gamma_0$ in $A(\\overline{F})$ is stable under the action of $\\Gamma$\non $A(\\overline{F})$. Let $\\lambda\\in\\Gamma$, then\n$$\n\\lambda(\\gamma_0)=\\lambda(g\\gamma g^{-1})=\\lambda(g)\\gamma\\lambda(g)^{-1},\n$$\nbecause $\\gamma\\in G(F)$. This computation shows that $\\lambda(\\gamma_0)$ is conjugate to\n$\\gamma_0$ by an element of $G(\\overline{F})$ and hence, by the argument above,\n$\\lambda(\\gamma_0)$ and $\\gamma_0$ lie in the same $W$-orbit, hence the image of $f$ lies\nin ${\\mathbb A}(F)$.\n\n\n\n\\subsection{The homomorphism $\\psi_G:\\Lambda_G\\longrightarrow W$}\\label{ss:psi-G}\nLet $\\Lambda_G=X_*(A)\/X_*(A_{sc})$, and let $A_G\\subseteq A$ be the identity\ncomponent of the center of $G$. We have the following natural maps:\n\\begin{equation}\\label{eq:l} \\Lambda_G\\twoheadrightarrow\\Lambda_G\/X_*(A_G)\\hookrightarrow P(R^{\\vee})\/Q(R^{\\vee}), \\end{equation}\nwhere $X_*(A_G)$ is identified with its image under the embedding\n$X_*(A_G)\\hookrightarrow X_*(A)\\twoheadrightarrow \\Lambda_G$ and $R^{\\vee}$ denotes the coroot system of\n$G$. Let $\\psi:P(R^{\\vee})\/Q(R^{\\vee})\\longrightarrow W$ be the map defined as in \\cite{b}\n(Chapter VI, \\S2, no.~3). (See also \\S\\ref{ss:psi} below.) We denote by\n$\\psi_G$ the composition\n$$\n\\Lambda_G\\longrightarrow P(R^{\\vee})\/Q(R^{\\vee})\\rarab{\\psi} W,\n$$\nwhere the first map is given by \\eqref{eq:l}.\n\n\\begin{rem} We note that $\\psi_G$ is the map $ps$ introduced\nin \\cite{k} (\\S 1.9, p. 9).\n\\end{rem}\n\n\n\n\\subsection{Newton stratification of ${\\mathbb A}(F)$} Let\n${\\mathfrak a}=X_*(A)\\otimes_{{\\mathbb Z}}{\\mathbb R}$ and let $P=MN$ be a parabolic subgroup of $G$\ncontaining $B$ (i.e., $P$ is a {\\em standard} parabolic subgroup) with  Levi\nsubgroup $M$ and  unipotent radical $N$. Let $W_M$ be the Weyl group of $M$\nidentified with a subgroup of $W$. Put ${\\mathfrak a}_M:=X_*(A_M)\\otimes_{{\\mathbb Z}}{\\mathbb R}$, where\nas above $A_M\\subseteq A$ denotes the identity component of the center of $M$.\nThen ${\\mathfrak a}_M$ can be identified with the set of fixed points of ${\\mathfrak a}$ under the\naction of $W_M$, and we have a surjection\n$$\np_M:{\\mathfrak a}\\twoheadrightarrow {\\mathfrak a}_M,\\quad x\\mapsto\\frac{1}{W_M}\\cdot\\sum_{w\\in W_M}w(x).\n$$\nSince $G_{sc}$ is simply connected, the derived group of $M$ is also simply\nconnected, which implies that $\\Lambda_M$ can be identified with the image of\n$X_*(A)$ under $p_M$. In what follows we write $\\Lambda_M\\hookrightarrow{\\mathfrak a}_M$ to indicate\nthat $\\Lambda_M$ is considered as  $p_M(X_*(A))\\subseteq{\\mathfrak a}_M$.\n\n\\medbreak\n\nLet ${\\mathcal N}_G$ be the subset of ${\\mathfrak a}$ defined as in \\cite{k} (\\S 1.3, p. 4). For the\nsake of completeness we repeat the facts about ${\\mathcal N}_G$ which will be used\nin this paper. First,\n$$\n{\\mathcal N}_G=\\coprod_{P}\\Lambda_P^+ \\subseteq{\\mathfrak a},\n$$\nwhere the union is taken over all standard parabolic subgroups of $G$ and for\neach such $P$ (in the above notation) $\\Lambda_P^+$ is a subset of\n$\\Lambda_M\\hookrightarrow{\\mathfrak a}_M$. Moreover, $\\Lambda_G^+=\\Lambda_G$, where $\\Lambda_G\\hookrightarrow{\\mathfrak a}_G$. Thus,\nfor each $\\nu\\in{\\mathcal N}_G$ there is a unique parabolic subgroup $P=MN$ such that\n$\\nu\\in\\Lambda_P^+$, hence the element $\\psi_M(\\nu)\\in W_M$ is defined and we put\n\\begin{equation}\\label{eq:we} w(\\nu):=\\psi_M(\\nu)\\in W.\\end{equation}\n\nAlso, for each $\\nu\\in{\\mathcal N}_G$ there is a certain non-empty irreducible subset\n${\\mathbb A}(F)_{\\nu}$ of ${\\mathbb A}(F)$ such that\n$$\n{\\mathbb A}(F)=\\coprod_{\\nu\\in{\\mathcal N}_G}{\\mathbb A}(F)_{\\nu}\n$$\n(\\cite{k}, Thm. 1.5.2, p. 6).\n\n\\medbreak\n\nIn what follows we will need one result about the sets ${\\mathbb A}(F)_{\\nu}$, which we\nstate as a lemma for future reference.\n\n\\begin{lem}\\label{l:kot}\nLet $\\nu\\in{\\mathcal N}_G$. Then $c\\in{\\mathbb A}(\\overline{F})$ belongs to ${\\mathbb A}(F)_{\\nu}$ if and only if\nthere is $a\\in A(\\overline{F})$ such that\n\\begin{eqnarray}\n&& c=c(a),\\label{eq:surj1}\\\\\n&&\\sigma(a)=g(a)\\quad\\text{for some }g\\in W,\\text{ and}\\label{eq:surj2}\\\\\n&& \\pair{\\lambda,\\nu}=\\operatorname{val}\\lambda(a)\\quad\\text{for any }\\lambda\\in X^*(A).\\label{eq:surj3}\n\\end{eqnarray}\n\\end{lem}\n\n\\begin{proof}\nSee Theorem 1.5.2(5) of \\cite{k}.\n\\end{proof}\n\n\\medbreak\n\nWe consider ${\\mathbb A}(F)_{\\nu}$ endowed with the topology induced from the topology\non ${\\mathbb A}(F)$ described in \\S\\ref{ss:base-field}. By a {\\em generic subset} of\n${\\mathbb A}(F)_{\\nu}$ we mean a subset containing an open dense subset of\n${\\mathbb A}(F)_{\\nu}$. Also, we say that a certain property {\\em holds for a generic\nelement of ${\\mathbb A}(F)_{\\nu}$} if it holds for all elements of some generic subset\nof ${\\mathbb A}(F)_{\\nu}$.\n\n\n\n\\subsection{Twists of a torus}\\label{ss:tor} Let $w\\in W$. In what follows we\nwill denote by $A^w$ the torus of $G$ obtained from $A$ by twisting the Galois\nstructure on $A$ by $w$. Then $X^*(A^w)=X^*(A)$ as abelian groups, and if\n$\\lambda\\in X^*(A^w)$, then the Galois action on $\\lambda$ is given by\n$$\n\\sigma\\lambda(a)=\\lambda(w^{-1}(a)), \\quad a\\in A.\n$$\nWe say that a torus $T$ of $G$ defined over $F$ is {\\em obtained from $A$ by\ntwisting by $w$} if and only if $X^*(T)\\cong X^*(A^w)$ as $\\Gamma$-modules (with\nthe natural action of $\\Gamma$ on $X^*(T)$).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Statement of the main theorem}\\label{s:main}\n\nWe keep all the notation introduced above. In this section we view $G$ and $A$\nas algebraic groups over $F$. Our goal is to state the main result mentioned in\nthe introduction (Theorem \\ref{t:1}), give a more precise reformulation of this\nresult (Theorem \\ref{t:2}), and reduce its proof to a special case (Theorem\n\\ref{t:3}).\n\n\n\\subsection{Main result} The next two sections, as well as \\S\\ref{ss:Conclusion},\nare devoted to the proof of\n\n\\begin{thm}\\label{t:1}\nLet $\\nu\\in{\\mathcal N}_G.$ Then for generic $c\\in{\\mathbb A}(F)_{\\nu}$  and a regular\nsemisimple element $\\gamma\\in G(F)$ such that $f(\\gamma)=c$ we have\n$$\nG_{\\gamma}\\cong A^{w(\\nu)},\n$$\nwhere $w(\\nu)$ is given by \\eqref{eq:we}.\n\\end{thm}\n\nLet us slightly reformulate Theorem \\ref{t:1}. As we explained in\n\\S\\ref{subsec:f}, there exists $g\\in G(\\overline{F})$ such that\n\\begin{eqnarray*}\ng\\gamma g^{-1}&=&\\gamma_0\\in A(\\overline{F})\\quad\\text{and}\\\\\ng\\sigma(g)^{-1}&\\in &N_G(A)(\\overline{F}).\n\\end{eqnarray*}\nLet $h=h(\\gamma)$ denote the element of $W$ corresponding to $g\\sigma(g)^{-1}$. Note\nthat $h$ is defined up to conjugation in $W$.\n\n\\begin{lem}\nWe have $G_{\\gamma}\\cong A^h.$\n\\end{lem}\n\\begin{proof} Since $gG_{\\gamma}g^{-1}=A$, conjugation by $g$ induces\nthe isomorphism $\\phi:X^*(A)\\longrightarrow X^*(G_{\\gamma})$. By \\S\\ref{ss:tor} it is enough\nto show that $\\phi$ commutes with the action of $\\sigma$. Equivalently,\n\\begin{equation}\\label{eq:gx} (\\sigma\\lambda)(gxg^{-1})=\\sigma\\big(\\phi(\\lambda)\\big)(x),\\quad\\text{for\nany }\\lambda\\in X^*(A),\\,x\\in G_{\\gamma}.\\end{equation} Since $A$ splits, the right side of\n\\eqref{eq:gx} is equal to $\\lambda\\big(\\sigma(g)x\\sigma(g)^{-1}\\big)$, and by the\ndefinition of the Galois action on $X^*(A^h)$, the left hand side of \\eqref{eq:gx} is equal to\n$$\\lambda\\big(h^{-1}(gxg^{-1})\\big)=\\lambda\\big(\\sigma(g)x\\sigma(g)^{-1}\\big).$$\n\\end{proof}\n\nAccording to this lemma to prove Theorem \\ref{t:1} it is enough to prove the\nfollowing\n\n\\begin{thm}\\label{t:2'}\nLet $\\nu\\in{\\mathcal N}_G$. There exists a generic subset $X_\\nu\\subset{\\mathbb A}(F)_\\nu$ such\nthat given $a\\in A(\\overline{F})$ for which $G_a$ is a torus and $c(a)\\in X_{\\nu}$, the\nelement $h\\in W$ given by \\begin{equation} \\sigma(a)=h(a)\\end{equation} is conjugate to $w(\\nu)$.\n\\end{thm}\n\n\n\n\n\n\n\\subsection{Construction of $X_\\nu$}\nRecall that $R\\subset X^*(A)$ denotes the root system of $G$, let $R^+\\subset\nR$ be the set of positive roots with respect to $B$, and let\n$$\n\\Omega:=\\prod_{\\alpha\\in R}\\big(1-e^{\\alpha}\\big)\\in {\\mathbb C}[X^*(A)].\n$$\nFor $\\nu\\in{\\mathcal N}_G$ denote\n$$\nm_{\\nu}:=\\sum_{\\alpha\\in R}\\max\\left\\{0,\\pair{\\alpha,\\nu}\\right\\}.\n$$\nObserve that $m_{\\nu}=\\pair{2\\rho,\\nu}$, where $$\\rho=\\frac{1}{2}\\cdot\\sum_\n{\\alpha\\in R^{+}}\\alpha\\in X^*(A).$$ (This follows from the facts mentioned in the\nfirst section of the proof of Lemma \\ref{l:en} below.) Also, if\n$\\nu\\in\\Lambda_G\\hookrightarrow{\\mathfrak a}_G$ then $m_{\\nu}=0$. Note that for any $a\\in A$ such that\n$c(a)\\in{\\mathbb A}(F)_{\\nu}$, we have $\\operatorname{val}\\Omega(a)\\leq m_{\\nu}$. Indeed, by Lemma\n\\ref{l:kot} there exists $b\\in A$ such that $c(a)=c(b)$ and\n$\\operatorname{val}\\lambda(b)=\\pair{\\lambda,\\nu}$ for any $\\lambda\\in X^*(A)$. Hence $a=g(b)$ for some\n$g\\in W$, and consequently\n$$\n\\operatorname{val}\\Omega(a)=\\operatorname{val}\\Omega(b)=\\sum_{\\alpha\\in R}\\operatorname{val}(1-\\alpha(b))\\leq\\sum_{\\alpha\\in\nR}\\max\\left\\{0,\\pair{\\alpha,\\nu}\\right\\}=m_{\\nu}.\n$$\n(Recall that with our conventions, $\\operatorname{val}$ is the negative of the usual valuation, so that $$\\operatorname{val}(a+b)\\leq\\max\\{\\operatorname{val} a,\\operatorname{val} b\\},\\quad\\text{for any }a,b\\in\\overline{F}.)$$\nOn the other hand, $\\Omega\\in{\\mathbb C}[X^*(A)]$ belongs to the subring ${\\mathbb C}[X^*(A)]^W$\nof $W$-invariant elements in ${\\mathbb C}[X^*(A)]$. Since $G$ comes from an algebraic\ngroup defined over ${\\mathbb C}$,\n$$\n{\\mathbb C}[X^*(A)]^W={\\mathbb C}[c_1,\\ldots,c_n,c_{l+1}^{-1},\\ldots,c_{n}^{-1}].\n$$\nTogether with the note above, this implies that\n$$\nX_{\\nu}:=\\big\\{c\\in{\\mathbb A}(F)_{\\nu}\\,\\big\\vert\\,\\exists\\, a\\in A\\text{ s.t. }\nc=c(a)\\text{ and }\\operatorname{val}\\Omega(a)=m_{\\nu}\\big\\}\n$$\nis an open subset of ${\\mathbb A}(F)_{\\nu}$.\n\n\\medbreak\n\nSince ${\\mathbb A}(F)_{\\nu}$ is irreducible, Theorem \\ref{t:2'} is a consequence of the\nfollowing result:\n\\begin{thm}\\label{t:2}\nFor each $\\nu\\in{\\mathcal N}_G$ the set $X_{\\nu}$ is non-empty and for any $a\\in A$ such\nthat $c(a)\\in X_{\\nu}$, the element $h\\in W$ given by \\begin{equation} \\sigma(a)=h(a) \\end{equation} is\nconjugate to $w(\\nu)$.\n\\end{thm}\n\n\\begin{rem}\\label{r:chert}\nObserve that the condition $c(a)\\in X_{\\nu}$ implies automatically that $G_a$\nis torus, or equivalently, $G_a=A$. Indeed, write $a=a_1a_2$, where $a_1\\in\nA_{sc}$ and $a_2\\in A_G$. Then $G_a=(G_{sc})_{a_1}\\cdot A_G$. Since $a_1\\in\nA_{sc}$, it is in particular a semisimple element of $G_{sc}$, and hence the\ncentralizer $(G_{sc})_{a_1}$ of $a_1$ in $G_{sc}$ is {\\em connected} by a deep\nresult of Springer and Steinberg (\\cite{ss}, Theorem II.3.9), which is false\nwithout the assumption that $G_{sc}$ is simply connected. Thus, $G_a$ is\nconnected, as a product of two connected algebraic groups. Now, assume that\n$G_a$ does not coincide with $A$. Then the Lie algebra $\\operatorname{Lie}(A)$ of $A$ is a\nproper subspace of the Lie algebra $\\operatorname{Lie}(G_a)$ of $G_a$. Moreover, $\\operatorname{Lie}(G_a)$\nis invariant under the adjoint action of $A$, since $A\\subset G_a$. This\nimplies that $\\operatorname{Lie}(G_a)$ contains a root subspace ${\\mathfrak g}_{\\alpha}$ for some $\\alpha\\in R$. Thus\n$\\alpha(a)=1$, and we get a contradiction with $c(a)\\in X_{\\nu}$.\n\\end{rem}\n\n\n\n\\subsection{Reduction from ${\\mathcal N}_G$ to $\\Lambda_G$}\nTo end this section, we observe that in Theorem \\ref{t:2} it is enough to\nassume that $\\nu\\in\\Lambda_G\\hookrightarrow{\\mathfrak a}_G$, which means that we only have to prove\n\n\\begin{thm}\\label{t:3}\nFor each $\\nu\\in\\Lambda_G\\hookrightarrow{\\mathfrak a}_G$ the set $X_{\\nu}$ is non-empty and for any\n$a\\in A$ such that $c(a)\\in X_{\\nu}$, the element $h\\in W$ given by \\begin{equation}\n\\sigma(a)=h(a) \\end{equation} is conjugate to $w(\\nu)=\\psi_G(\\nu)$.\n\\end{thm}\n\nIndeed, we have the following result.\n\n\\begin{lem}\\label{l:en}\nTheorem $\\ref{t:3}$ implies Theorem $\\ref{t:2}.$\n\\end{lem}\n\n\\begin{proof}\nLet $\\nu\\in\\Lambda_P^+$ for some standard parabolic subgroup $P$ of $G$ with the\nLevi subgroup $M$ containing $A$, so that $\\Lambda_P^{+}\\subseteq\\Lambda_M\\hookrightarrow{\\mathfrak a}_M$.\nDenote by $\\Delta_M\\subseteq\\Delta$ the set of simple roots of $M$. Also, let\n$R_M\\subseteq R$ denote the root system of $M$ and let $$\\Omega_M:=\\prod_{\\alpha\\in\nR_M}(1-e^{\\alpha})\\in{\\mathbb C}[X^*(A)].$$ Recall (\\cite{k}, p. 4) that $\\Lambda_P^+$ is contained in\nthe set of elements\n $x\\in{\\mathfrak a}_M$ such that $\\pair{\\alpha,x}>0$ for any $\\alpha\\in\\Delta\\backslash\\Delta_M$.\nSince\n$${\\mathfrak a}_M=\\big\\{x\\in{\\mathfrak a}\\,\\big\\vert\\,\\pair{\\alpha,x}=0,\\,\\forall\\alpha\\in \\Delta_M\\big\\},\n$$\nwe conclude that $\\pair{\\alpha,\\nu}=0$ for any $\\alpha\\in R_M$ and $\\pair{\\alpha,\\nu}\\ne\n0$ for any $\\alpha\\in R\\backslash R_M$. Thus, if $a\\in A$ satisfies\n\\eqref{eq:surj3} and $c(a)\\in{\\mathbb A}(F)_{\\nu}$, then\n\\begin{equation}\\label{eq:op}\\operatorname{val}\\Omega(a)=m_{\\nu}\\Longleftrightarrow\\operatorname{val}\\Omega_M(a)=0.\\end{equation}\n\n\\medbreak\n\nLet ${\\mathbb A}_M$ and $c_M:A\/W_M\\rarab{\\sim}{\\mathbb A}_M$ denote respectively the set ${\\mathbb A}$\nand the map $c$ corresponding to $M$. Consider the map from ${\\mathbb A}_M$ to ${\\mathbb A}$\ndefined by the composition of $c_M^{-1}$ with the natural map $A\/W_M\\longrightarrow A\/W$\n(induced by the embedding $W_M\\hookrightarrow W$) followed by $c$. It is easy\nto check that this map induces a map $\\pi_M:{\\mathbb A}_M(F)_{\\nu}\\longrightarrow{\\mathbb A}(F)_{\\nu}$. We\nare now ready to prove the lemma. Let us show first that $X_{\\nu}$ is\nnon-empty. Let $X_{\\nu}(M)$ denote the set $X_{\\nu}$ defined for $M$. By\nTheorem \\ref{t:3} and Lemma \\ref{l:kot}, there exists $a\\in A$ satisfying\n\\eqref{eq:surj3} such that $c_M(a)\\in{\\mathbb A}_M(F)_{\\nu}$ and $\\operatorname{val}\\Omega_M(a)=0$. Then\n$c(a)=\\pi_M(c_M(a))\\in{\\mathbb A}(F)_{\\nu}$ and $\\operatorname{val}\\Omega(a)=m_{\\nu}$ by \\eqref{eq:op}.\nThus, $c(a)\\in X_{\\nu}$.\n\n\\medbreak\n\nNow we will show that Theorem \\ref{t:2} holds for $X_{\\nu}$. Let $a\\in A$\nsatisfy $c(a)\\in X_{\\nu}$ and $\\sigma(a)=h(a)$. Then by Lemma \\ref{l:kot} there\nexists $b\\in A$ satisfying  \\eqref{eq:surj3} such that $c(a)=c(b)$, hence\n$b=s(a)$ for some $s\\in W$. Thus, $\\sigma(b)=g(b)$ for some $g\\in W$. Since $a$ (and hence $b$) is regular semisimple, as it follows\nfrom Remark \\ref{r:chert}, we see that $g$ is conjugate to $h$. Hence it is enough to\nshow that $g$ is conjugate to $w(\\nu)=\\psi_M(\\nu)$. For any $\\lambda\\in X^*(A)$ we\nhave\n$$\n\\pair{\\lambda,\\nu}=\\operatorname{val}\\lambda(b)=\\operatorname{val}\\sigma(\\lambda(b))=\\operatorname{val} g^{-1}\\lambda(b)=\\pair{\\lambda,g\\nu},\n$$\nwhich implies $\\nu=g\\nu$. Thus, $g\\in W_M$ by standard facts about reflection\ngroups, cf. \\cite{b}. By Lemma \\ref{l:kot} this gives $c_M(b)\\in{\\mathbb A}_M(F)_{\\nu}$\nand also $\\operatorname{val}\\Omega_M(b)=0$ by \\eqref{eq:op}. Hence, $c_M(b)\\in X_{\\nu}(M)$ and\n$g$ is conjugate to $\\psi_M(\\nu)$ by Theorem \\ref{t:3}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A result about root systems}\\label{s:systems}\n\n\n\nThe goal of this section is to show that Theorem \\ref{t:3} follows from a\ncertain statement about root systems (Proposition \\ref{p:sys} below). This\nprovides a link between our main result (which is specific to algebraic groups\nover $F$) and the theory of semisimple algebraic groups over ${\\mathbb C}$. Note that\nthe torus $T$ appearing in \\S\\ref{ss:generalities} corresponds to the torus\n$A_{sc}$ defined in \\S\\ref{ss:setup}; otherwise our notation remains the same\nas before.\n\n\n\n\\subsection{Generalities}\\label{ss:generalities}\nLet $R$ be a (reduced) root system in a real vector space $V$ and let $T$ be\nthe complex torus with character lattice $P(R)$, i.e.,\n$$\nT=\\operatorname{Spec}{\\mathbb C}[P(R)].\n$$\nFor each $\\mu\\in P(R^{\\vee})$ we let $x_{\\mu}$ denote the composition\n$$\nP(R)\\rarab{\\pair{\\cdot,\\mu}}{\\mathbb Q}\/{\\mathbb Z}\\hrarab{\\exp(-2\\pi\n\\sqrt{-1}\\cdot)}{\\mathbb C}^{\\times}.\n$$\nThus $x_{\\mu}$ is a group homomorphism from $P(R)$ to ${\\mathbb C}^{\\times}$, and we\nidentify it with the (complex) point of $T$ it defines.\n\n\\medbreak\n\nLet $W$ denote the Weyl group of $R$, acting on $P(R)$ and hence on $T$ in  the\nusual way, and let\n$$\n\\psi:P(R^{\\vee})\/Q(R^{\\vee})\\longrightarrow W\n$$\ndenote the homomorphism mentioned in \\S\\ref{ss:psi-G} above (see also \\S\\ref{ss:psi}).\n\n\\medbreak\n\nRecall that an element $u\\in T$ is said to be {\\em regular} if the stabilizer\nof $u$ in $W$ is trivial, or, equivalently, if\n$$\n\\alpha(u)\\ne 1,\\quad\\forall\\alpha\\in R.\n$$\n(The equivalence of the two properties is not an obvious statement, and follows\nfrom the same argument as in Remark \\ref{r:chert}.)\n\n\n\n\n\\subsection{Reduction of Theorem \\ref{t:3} to Proposition \\ref{p:sys}}\\label{ss:reduction}\nIn this section we assume\n\n\\begin{prop}\\label{p:sys}\nLet $\\mu\\in P(R^{\\vee})$ and let $h\\in W$. There exists a regular element\n$u\\in T$ with $h(u)=x_{\\mu} u$ if and only if $h$ is conjugate to\n$\\psi(\\bar{\\mu})$.\n\\end{prop}\n\nThe proof is given in \\S\\ref{ss:Conclusion}.\n\n\\begin{lem}\\label{l:enough}\nProposition $\\ref{p:sys}$ implies Theorem $\\ref{t:3}.$\n\\end{lem}\n\n\\begin{proof}\nIn the notation of Theorem \\ref{t:3}, let $R$ be the root system of $G$ with\nrespect to $A$, and note that the torus $T=\\operatorname{Spec}{\\mathbb C}[P(R)]$ is identified with the maximal\ntorus $A_{sc}$ of $G_{sc}$. Fix $\\nu\\in \\Lambda_G$. We first show that $X_{\\nu}$ is\nnon-empty. Put\n$$\nm_i:=\\pair{\\omega_i,\\nu}\\in{\\mathbb Q},\\quad i\\in\\{1,\\ldots,n\\},\n$$\nwhere as usual we consider $\\nu$ as an element of ${\\mathfrak a}_G$ under the embedding\n$\\Lambda_G\\hookrightarrow{\\mathfrak a}_G$. Let $\\mu\\in P(R^{\\vee})$ be such that\n$\\nu\\mapsto\\bar{\\mu}$ under the map\n$$\n\\Lambda_G\\twoheadrightarrow\\Lambda_G\/X_*(A_G)\\hookrightarrow P(R^{\\vee})\/Q(R^{\\vee}).\n$$\nThen \\begin{equation}\\label{eq:equiv}\n\\pair{\\omega_i,\\nu}\\equiv-\\pair{\\varpi_i,\\mu}\\mod\\,{\\mathbb Z}\\quad(1\\leq i\\leq l) \\end{equation}\n(\\cite{k}, p. 14). Furthermore, by definition $\\psi_G(\\nu)=\\psi(\\bar{\\mu})$. By\nProposition \\ref{p:sys} applied to $R$ and $\\mu$ there exists $g\\in W$ and a\nregular element $u\\in T$ such that $g(u)=x_{\\mu} u$. Note that in view of\n\\eqref{eq:equiv} we have $g(u)=x_{\\mu} u$ if and only if \\begin{equation}\\label{eq:starr}\ng^{-1}\\varpi_i(u)=\\exp\\left(2\\pi\\sqrt{-1}m_i\\right)\\cdot\\varpi_i(u),\\quad\n\\forall i\\in\\{1,\\ldots,l\\}. \\end{equation} Let $a\\in A$ be given by\n\\begin{eqnarray*}\n\\omega_i(a)&=&\\epsilon^{m_i}\\cdot\\varpi_i(u),\\quad i\\in\\{1,\\ldots,l\\},\\\\\n\\omega_i(a)&=&\\epsilon^{m_i},\\,\\,\\,\\,\\,\\,\\,\\quad\\quad\\quad i\\in\\{l+1,\\ldots,n\\}.\n\\end{eqnarray*}\nWe claim that $c(a)\\in X_{\\nu}$. Recall that $c\\in{\\mathbb A}(\\overline{F})$ belongs to\n$X_{\\nu}$ if and only if there is $b\\in A$ such that\n\\begin{eqnarray}\n c & = & c(b),\\label{eq:surj10}\\\\\n \\pair{\\omega_i,\\nu} & = & \\operatorname{val}\\omega_i(b),\\quad\\forall i\\in \\{1,\\ldots,n\\},\\label{eq:surj20}\\\\\n\\sigma(b) & = & s(b)\\,\\text{ for some }s\\in W,\\label{eq:surj30}\\text{\nand}\\\\\n\\operatorname{val}\\Omega(b) & = & 0\\label{eq:surj40}.\n\\end{eqnarray}\nClearly, \\eqref{eq:surj20} holds for $b=a$. For \\eqref{eq:surj40}, note that\n$$\n\\alpha(a)=\\epsilon^{\\pair{\\alpha,\\nu}}\\cdot\\alpha(u)=\\alpha(u),\\quad\\forall\\alpha\\in R,\n$$\nwhere on the right we consider $\\alpha$ as a character of $A_{sc}$. Since $u$ is\nregular, this gives \\eqref{eq:surj40} for $b=a$. Thus, it is enough to show\nthat $\\sigma(a)=g(a)$. Since $m_{l+1},\\ldots,m_n\\in{\\mathbb Z}$ and $W$ acts trivially on\n$\\omega_{l+1},\\ldots,\\omega_n$, this is equivalent to \\begin{equation}\\label{eq:**}\n\\sigma(\\omega_i(a))=g^{-1}\\omega_i(a),\\quad\\forall i\\in\\{1,\\ldots,l\\}.\\end{equation} For each $i$\nusing \\eqref{eq:starr} we have\n$$\n\\sigma(\\omega_i(a))=\\epsilon^{m_i}\\cdot\\exp\\left(2\\pi\\sqrt{-1}m_i\\right)\\cdot\\varpi_i(u)=\\epsilon^{m_i}\\cdot\ng^{-1}\\varpi_i(u)=g^{-1}\\omega_i(a),\n$$\nwhich proves \\eqref{eq:**}.\n\n\\medbreak\n\nLet us show now that Theorem \\ref{t:3} holds for $X_{\\nu}$. Let $a\\in A$ and\n$h\\in W$ be such that $c(a)\\in X_{\\nu}$ and $\\sigma(a)=h(a)$. As was explained in\nthe last paragraph of the proof of Lemma \\ref{l:en}, without loss of generality we\ncan assume that $\\operatorname{val}\\lambda(a)=\\pair{\\lambda,\\nu}$ for any $\\lambda\\in X^*(A)$. We need to\nshow that $h$ is conjugate to $w(\\nu)=\\psi_G(\\nu)$. Note first that there\nexists an element $z$ of the center $Z$ of $G$ such that\n\\begin{eqnarray*}\n&& \\omega_i(za)=\\epsilon^{m_i}\\cdot(v_i+\\dotsb),\\quad v_i\\in{\\mathbb C}^{\\times},\\,\\, i\\in\\{1,\\ldots,n\\},\\\\\n&& v_{l+1}=\\dotsb=v_n=1.\n\\end{eqnarray*}\nHere the dots in the formula for $\\omega_i(za)$ denote an element in the maximal ideal of the valuation ring in $\\overline{F}$. Let $u\\in T=\\operatorname{Spec}{\\mathbb C}[P(R)]$ be given by\n$$\n\\varpi_i(u)=v_i,\\quad 1\\leq i\\leq l.\n$$\nWe claim that $u$ is regular and \\begin{equation}\\label{eq:***} h(u)=x_{\\mu} u,\\end{equation}\nwhich together with Proposition \\ref{p:sys} and the fact that $\\psi_G(\\nu)=\\psi\n(\\bar{\\mu})$ implies Theorem \\ref{t:3}.\n\n\\medbreak\n\nSince $z\\in Z$, we have\n$$\n\\alpha(a)=\\alpha(za)=\\alpha(u)+\\dotsb,\\quad \\forall\\alpha\\in R,\n$$\nwhere the dots denote elements in the maximal ideal of the valuation ring in $\\overline{F}$. Since\n$\\operatorname{val}\\Omega(a)=0$, we conclude that $\\alpha(u)\\ne 1$ for any $\\alpha\\in R$, i.e., $u$ is\nregular. Furthermore,\nfor each $i\\in\\{1,\\ldots,l\\}$ we get\n\\begin{eqnarray*}\nh^{-1}\\omega_i(za)&=&\\epsilon^{m_i}\\cdot(h^{-1}\\varpi_i(u)+\\dotsb),\\text{ and}\\\\\n\\sigma(\\omega_i(za))&=&\\epsilon^{m_i}\\cdot\\exp(2\\pi\\sqrt{-1}m_i)\\cdot(\\varpi_i(u)+\\dotsb),\n\\end{eqnarray*}\nwhere as usual the dots denote elements in the maximal ideal of the valuation ring in $\\overline{F}$. Since\n$$\n\\sigma(za)=h(za)\\Longleftrightarrow \\sigma(a)=h(a),\n$$ we obtain\n$$\nh^{-1}\\varpi_i(u)=\\exp(2\\pi\\sqrt{-1}m_i)\\cdot\\varpi_i(u),\\quad\\forall\ni\\in\\{1,\\ldots,l\\},\n$$\nwhich is equivalent to \\eqref{eq:***} by \\eqref{eq:starr}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Springer's theory for the torus $A$}\\label{s:Springer}\n\n\nThis section (which contains analogues of some results of Springer \\cite{s})\nand the next one can be read independently of the rest of the paper, and are\ninteresting in their own right. The main results are Theorem \\ref{t:abst} and\nTheorem \\ref{t:s}. The reader who is only interested in Newton stratifications\nmay decide to take the statement of Theorem \\ref{t:s} on faith and skip\ndirectly to \\S\\ref{ss:Conclusion}, which contains a proof of Proposition\n\\ref{p:sys}.\n\n\n\\subsection{General conjugacy theorem}\\label{ss:abstract}\nLet $X$ be a separated algebraic variety over ${\\mathbb C}$, and let $Z$ and $W$ be finite\ngroups acting on $X$ by morphisms. For $z\\in Z$ (resp., $w\\in W$) and $x\\in X$\nwe denote by $z\\cdot x$ (resp., $w(x)$) the action of $z$ (resp., of $w$) on\n$x$. Assume that the actions of $Z$ and $W$ commute, i.e.,\n$$\nw(z\\cdot x)=z\\cdot w(x),\\quad\\forall w\\in W,\\,z\\in Z,\\,x\\in X.\n$$\nDenote $X^\\circ:=\\left\\{x\\in X\\vert\\,W_x=\\{1\\}\\right\\}$. Let $Y=X\/W$ be the\nquotient considered as a topological space with the quotient topology of the Zariski topology on $X$. Since\nthe actions of $Z$ and $W$ on $X$ commute, we have the induced action of $Z$ on\n$Y$.\n\n\\begin{thm}\\label{t:abst}\nLet $z\\in Z$ be such that $$Y^z:=\\left\\{y\\in Y\\vert\\,z\\cdot y=y\\right\\}$$ is\nirreducible. If there exist $w_1,w_2\\in W$ and $x_1,x_2\\in X^\\circ$ such that\n$$\nw_i(x_i)=z\\cdot x_i,\\quad i=1,2,\n$$\nthen $w_1$ is conjugate to $w_2$.\n\\end{thm}\n\\begin{proof}\nLet $f:X\\twoheadrightarrow Y$ denote the quotient map. First, note that without loss of\ngenerality we can assume that $X=X^\\circ$. Indeed, $X^\\circ$ is an open, $W$- and\n$Z$-invariant subset of $X$. In particular, $X^\\circ$ is separated. Also, by\nassumption, $f(X^\\circ)^z$ is nonempty and it is an open subset of $Y^z$, because\n$f$ is open and $f(X^\\circ)^z=f(X^\\circ)\\cap Y^z$. Thus, $f(X^\\circ)^z$ is irreducible.\n\n\\smallbreak\n\nThus assume that $X=X^\\circ$. For $w\\in W$ denote $$X(w,z):=\\left\\{x\\in X\\vert\\,w(x)=z\\cdot\nx\\right\\}.$$ Since $X$ is separated, each $X(w,z)$ is closed. Also, for $w\\ne\nw'\\in W$ we have $X(w,z)\\cap X(w',z)=\\emptyset$. Therefore\n$$\nf^{-1}(Y^z)=\\coprod_{w\\in W}X(w,z),\n$$\nand each $X(w,z)$ is open in $f^{-1}(Y^z)$. Since the restriction of $f$ to\n$f^{-1}(Y^z)$ is also an open map, we see that $f\\left(X(w_1,z)\\right)$ and\n$f\\left(X(w_2,z)\\right)$ intersect, being nonempty open subsets of the irreducible space\n$Y^z$. Hence there exist $u,v\\in X$ and $g\\in W$ such that\n$u=g(v)$, $w_1(u)=z\\cdot u$, and $w_2(v)=z\\cdot v$. Combining these three\nequalities and taking into account that $X=X^\\circ$, we get $w_1=gw_2g^{-1}$.\n\\end{proof}\n\n\n\\begin{rem}\nThis argument was obtained by a careful analysis of the proof of Theorem 4.2(iv) in \\cite{s}.\n\\end{rem}\n\n\n\\begin{cor}\\label{c:dim}\nSuppose $X$ is a quasi-projective variety $($so that $Y$ has the structure of an\nalgebraic variety where the underlying topology coincides with the quotient topology$)$, and\n$z\\in Z$ is such that $Y^z$ is irreducible. If $X^\\circ\\cap\nX(w,z)\\ne\\emptyset$ for some $w\\in W$, then $\\operatorname{dim} X(w,z)=\\operatorname{dim} Y^z$. Conversely,\nassume that $Y^z\\cap f(X^\\circ)\\ne\\emptyset$, and let $w\\in W$. If $C$ is an irreducible\ncomponent of $X(w,z)$ with $\\operatorname{dim} C=\\operatorname{dim} Y^z$, then $X^\\circ\\cap C\\ne\\emptyset$.\n\\end{cor}\n\n\\begin{proof}\nLet  $X^\\circ\\cap X(w,z)\\ne\\emptyset$. Since $f$ is open and finite, we have\n\\[\n\\operatorname{dim} X(w,z) \\geq \\operatorname{dim}\n\\left(X^\\circ\\cap X(w,z)\\right)=\\operatorname{dim} f(X^\\circ\\cap X(w,z))=\\operatorname{dim} Y^z \\geq \\operatorname{dim} X(w,z),\n\\]\nwhere the second equality holds because $f(X^\\circ\\cap X(w,z))$ is nonempty and open in $Y^z$.\n\n\\smallbreak\n\nLet $\\operatorname{dim} C=\\operatorname{dim} Y^z$. Then $f(C)$ is dense in $Y^z$, hence intersects $f(X^\\circ)$\nnontrivially, which implies $X^\\circ\\cap C\\ne\\emptyset$, because $X^\\circ$ is\n$W$-invariant.\n\\end{proof}\n\n\n\\subsection{New conventions}\\label{ss:Conventions} In the rest of the paper we will\nbe working in a special case of the setup described in \\S\\ref{ss:setup}. Let\n$G$ be a connected and simply connected simple algebraic group over ${\\mathbb C}$; thus\n$G_{sc}=G$ and $A_{sc}=A$. All our algebraic varieties (resp., groups) will be\ndefined over ${\\mathbb C}$, and will be implicitly identified with their sets (resp.,\ngroups) of complex points; in particular, $G$, $A$, $B$ will stand for\n$G({\\mathbb C})$, $A({\\mathbb C})$, $B({\\mathbb C})$, respectively. The tangent space to an algebraic\nvariety $Y$ at a point $y\\in Y$ will be denoted by $T_y Y$, and if $f:Y\\longrightarrow Y'$\nis a morphism to another algebraic variety $Y'$, its differential at $y$ will\nbe denoted by $D_y f : T_y Y \\longrightarrow T_{f(y)} Y'$.\n\n\n\n\n\\subsection{A variation on Springer's theory of regular elements}\\label{ss:springer}\nThe Weyl group $W=W(G,A)$ acts naturally on $A$ and on the Lie algebra\n$\\operatorname{Lie}(A)$ of $A$. It is well known that $W$ is a finite complex reflection\ngroup in $\\operatorname{Lie}(A)$; thus Springer's results \\cite{s} apply to it. In\nparticular, we recall that Springer introduces the notion of a {\\em regular\nelement} of $W$, namely, it is an element $g\\in W$ which has a regular\neigenvector $v\\in\\operatorname{Lie}(A)$ (the order of the corresponding eigenvalue is then\nnecessarily equal to the order of $g$). Further, he proves a number of useful\nand nontrivial results about regular elements; this includes the fact that if\nthere exists a regular element $g\\in W$ of order $d\\in{\\mathbb N}$, then all such\nelements form a single conjugacy class ({\\em op.~cit.}, Theorem 4.2(iv)), as\nwell as an explicit determination of the eigenvalues of a regular element ({\\em\nop.~cit.}, Theorem 4.2(v)).\n\n\\medbreak\n\nIf one tries to find a version of Springer's theory for the action of $W$ on\nthe torus $A$ itself, rather than on $\\operatorname{Lie}(A)$, the first problem one\nencounters is how to define the analogue of an eigenvector. Indeed, if $\\zeta$ is\na root of unity in ${\\mathbb C}$, the equation $g(v)=\\zeta v$ makes no sense in $A$. Thus\none is led to trying to replace the group of roots of unity by an abelian group\nacting on $A$ in a way which commutes with the Weyl group action. We choose the\nmost naive answer (in some sense), yet one which leads to nontrivial results\nthat have several interesting applications, as the present paper already\ndemonstrates. Namely, the group of roots of unity will be replaced by $Z$, the\ncenter of $G$, acting on $A$ by multiplication. We recall that $Z$ is a finite\nabelian group which is non-canonically isomorphic to $P(R^\\vee)\/Q(R^\\vee)$, and\nwe recall the homomorphism $\\psi:P(R^\\vee)\/Q(R^\\vee)\\longrightarrow W$ used earlier in the\npaper (its definition is reviewed in \\S\\ref{ss:psi} below).\n\n\\medbreak\n\nOur main result in this setup is the following\n\n\\begin{thm}\\label{t:s}\n\\begin{enumerate}[$(a)$]\n\\item For every $x\\in Z$, there exists $g\\in W$ for which there is\na regular element $u\\in A$ satisfying $g(u)=x u$. Moreover, if we choose an\nembedding of groups ${\\mathbb Q}\/{\\mathbb Z}\\hookrightarrow\\bC^\\times$, which induces an isomorphism $Z\\cong\nP(R^\\vee)\/Q(R^\\vee)$, and denote by $\\mu_x\\in P(R^\\vee)\/Q(R^\\vee)$ the element\ncorresponding to $x$ under this isomorphism, then one can take $g=\\psi(\\mu_x)$.\n\\item If $x\\in Z$ is fixed, the element $g$ satisfying the property above is\ndetermined uniquely up to conjugation.\n\\item The eigenvalues of such an element $g$ acting on\n$\\operatorname{Lie}(A)$, counting the multiplicities, are precisely the complex numbers\n$\\{\\varpi_i(x)^{-1}\\}_{i\\in I}$.\n\\end{enumerate}\n\\end{thm}\n\nPart (a) of this theorem follows from a stronger result (Theorem\n\\ref{t:monodromy}) proved in Section \\ref{s:geometric}. Namely, it is possible\nto choose {\\em one} regular element $u\\in A$ that works for all $x\\in Z$\nsimultaneously. The rest of the section is devoted to the proofs of parts (b)\nand (c).\n\n\n\n\\subsection{Proof of Theorem \\ref{t:s}(b)}\\label{ss:Springer} Let $l$ denote the\nrank of $G$ and for $x\\in Z$ let $a(x)$ denote the number of elements in the\nset $\\{ 1\\leq i\\leq l \\,\\bigl\\lvert\\, \\varpi_i(x)=1 \\}$. Also, as in\n\\S\\ref{ss:abstract}, we introduce, for each $g\\in W$ and $x\\in Z$, the subset\n\\[\nA(g,x) = \\{ u\\in A \\,\\bigl\\lvert\\, g(u)=x u \\}.\n\\]\nClearly, Theorem \\ref{t:s}(b) follows from the following more\nprecise proposition, which is an analogue of Theorem 4.2(iv) in \\cite{s}:\n\\begin{prop}\nIf $g\\in W$, $x\\in Z$, the following conditions are equivalent:\n\\begin{enumerate}[$(i)$]\n\\item $\\operatorname{dim} A(g,x)=a(x)$;\n\\item there exists a regular element $u\\in A(g,x)$;\n\\item there exists a regular element in every connected component of $A(g,x)$.\n\\end{enumerate}\nMoreover, for fixed $x$, the elements of $W$ satisfying these properties form a\nsingle conjugacy class.\n\\end{prop}\n\n\\begin{proof}\nNote that although $A(g,x)$ may be disconnected, it is clearly a torsor for the\nsubgroup $A^g$ consisting of the elements of $A$ that are fixed by $g$. In\nparticular, $A(g,x)$ is smooth, its connected components ($=$irreducible\ncomponents) correspond to the connected components of $A^g$, and consequently\nhave the same dimension.\n\n\\smallbreak\n\nWe recall from \\S\\ref{ss:adj-quotient} that we have a $W$-invariant polynomial\nmorphism $c:A\\longrightarrow{\\mathbb A}={\\mathbb A}^l$ whose coordinates $c_i$ are given by\n\\[\nc_i(u) = \\sum_{\\lambda\\in W\\varpi_i} e^\\lambda(u),\\quad u\\in A,\n\\]\nand that $c$ identifies ${\\mathbb A}$ with the quotient $A\/W$. We apply Theorem\n\\ref{t:abst} and Corollary \\ref{c:dim} to $A$ with the action of $Z$ by\nmultiplication and the natural action of $W$. We only need to show that ${\\mathbb A}^x$\nis irreducible. We have\n$$\n{\\mathbb A}^x=\\left\\{(c_i)\\in{\\mathbb A} \\,\\,\\bigl\\lvert\\,\\, c_i=\\varpi_i(x)\\cdot c_i,\\,\\,\n1\\leq i\\leq l \\right\\},\n$$\nwhich implies that ${\\mathbb A}^x$ is an affine space of dimension $a(x)$. Thus, the\nassumptions of Theorem \\ref{t:abst} are satisfied. Using Theorem \\ref{t:s}(a),\nwe see that Corollary \\ref{c:dim} can be applied to prove the equivalence of (i) -- (iii), and\nthe proposition follows from these two results.\n\\end{proof}\n\n\n\n\n\n\\begin{rem}\nIf $G$ is a finite reflection group in a complex vector space $V$, the quotient\n$V\/G$ admits a description similar to the description of $A\/W$ used above.\nUsing this description one can prove Theorem 4.2(iv) of \\cite{s} in a way which\nis completely analogous to the last proof. In the notation of\n\\S\\ref{ss:abstract}, the role of $X$ is played by $V$, the role of $W$ is\nplayed by $G$, and the role of $Z$ is played by the group of roots of unity of\na fixed order $d\\in{\\mathbb N}$. In fact, this argument is essentially identical to the\none used by Springer, modulo some elementary simplifications.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{t:s}(c)}\\label{ss:Eigenvalues} This subsection is\nan imitation of the proof of Theorem 4.2(v) in \\cite{s}. However, we give a\ncoordinate-free version of Springer's argument.\n\n\\smallbreak\n\nNote that the map $c:A\\longrightarrow{\\mathbb A}$ has the property that the differential $D_u\nc:T_u A \\longrightarrow T_{c(u)}{\\mathbb A}$ is an isomorphism if and only if $u\\in A$ is regular.\nFor each $x\\in Z$, we define $m_x:A\\longrightarrow A$ to be the map $u\\mapsto xu$; since\n$x$ is $W$-invariant, it is clear that we have a commutative diagram\n\\begin{equation}\\label{e:Dilation}\n\\xymatrix{\n  A \\ar[dd]_c \\ar[rr]^{m_x}   &    &   A \\ar[dd]^c \\\\\n    &     &     \\\\\n  {\\mathbb A}  \\ar[rr]^{\\overline{m}_x} &    &   {\\mathbb A}\n   }\n\\end{equation}\nwhere $\\overline{m}_x:{\\mathbb A}\\longrightarrow{\\mathbb A}$ is defined by $(a_i)_{i\\in I}\\mapsto (\\varpi_i(x)\\cdot\na_i)_{i\\in I}$. In particular, given an element $u\\in A$, we have (imitating\n\\cite{s}, \\S3):\n\\[\nxu=g(u)\\quad\\text{for some  } g\\in W \\iff c(xu)=c(u) \\iff \\overline{m}_x(c(u))=c(u)\n\\]\n\\[\n\\iff \\text{for each } i\\in I, \\text{ either } c_i(u)=0, \\text{ or } \\varpi_i(x)=1.\n\\]\n\n\\smallbreak\n\nLet $g\\in G$ and $x\\in Z$ be fixed, and let us also fix a regular element $u\\in\nA(g,x)$ (assuming it exists). The diagram \\eqref{e:Dilation} and the fact that\n$c\\circ g=c$ yield two commutative diagrams (note that $c(xu)=c(g(u))=c(u)$):\n\\[\n\\xymatrix{\n  T_u A \\ar[ddr]_{D_u c} \\ar[rr]^{D_u g}   &    &   T_{xu}A \\ar[ddl]^{D_{xu}c} \\\\\n    &     &     \\\\\n   &     T_{c(u)}{\\mathbb A}     &\n   }\n  \\qquad\\text{and}\\qquad\n\\xymatrix{\n  T_u A \\ar[dd]_{D_u c} \\ar[rr]^{D_u m_x}   &    &   T_{xu}A \\ar[dd]^{D_{xu}c} \\\\\n    &     &     \\\\\n  T_{c(u)} {\\mathbb A}  \\ar[rr]^{D_{c(u)}\\overline{m}_x} &    &  T_{c(u)} {\\mathbb A}\n   }\n\\]\nSince $u$, and hence $xu$, are regular, the maps $D_u c$ and $D_{xu} c$ are\nisomorphisms. Therefore the automorphism\n\\[\n(D_u g)^{-1}\\circ D_u m_x : T_u A \\rarab{\\simeq} T_u A\n\\]\nis equal to\n\\begin{eqnarray*}\n(D_u g)^{-1}\\circ D_u m_x &=& (D_u c)^{-1} \\circ (D_{xu} c) \\circ (D_{xu} c)^{-1} \\circ (D_{c(u)}\\overline{m}_x) \\circ D_u c \\\\\n &=& (D_u c)^{-1} \\circ (D_{c(u)}\\overline{m}_x) \\circ (D_u c).\n\\end{eqnarray*}\nIn particular, $(D_u g)^{-1}\\circ D_u m_x$ has the same eigenvalues as\n$D_{c(u)} \\overline{m}_x$. But $D_{c(u)} \\overline{m}_x$ is obviously given by a diagonal matrix\nin the standard basis, whose eigenvalues are the numbers $\\varpi_i(x)$. On the\nother hand, the chain rule yields\n\\[\n(D_u g)^{-1}\\circ D_u m_x = D_u (g^{-1}\\circ m_x) = D_u(m_x\\circ g^{-1}) =\n(D_{g^{-1}(u)} m_x)\\circ D_u(g^{-1}),\n\\]\nand $(D_{g^{-1}(u)} m_x)\\circ D_u(g^{-1})$ has the same eigenvalues as\n\\[\n(D_1 m_u)^{-1} \\circ (D_{g^{-1}(u)} m_x) \\circ D_u(g^{-1}) \\circ D_1 m_u = D_1(\nm_u^{-1}\\circ m_x \\circ g^{-1} \\circ m_u) = D_1(g^{-1}).\n\\]\nBut $D_1(g^{-1})$, the differential of $g^{-1}$ at the identity element $1\\in\nA$, is nothing but the automorphism by which $g^{-1}\\in W$ acts on $\\operatorname{Lie}(A)$.\nHence we have shown that the eigenvalues of $g^{-1}$ are the numbers\n$\\varpi_i(x)$, completing the proof of part (c) of Theorem \\ref{t:s}.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Proposition \\ref{p:sys}}\\label{ss:Conclusion} It is clear that\nif Proposition \\ref{p:sys} holds for two root systems, then it also holds for\ntheir direct sum; thus it suffices to prove it for a reduced and irreducible\nroot system. It is then obvious that parts (a) and (b) of Theorem \\ref{t:s} imply\nthe desired result, which finally completes the proof of Theorem \\ref{t:3}, and hence of Theorem \\ref{t:1}.\n\n\\begin{rem}\\label{r:kottwitz}\nAs an additional bonus, we note that Theorem \\ref{t:s} yields an alternate\nproof of Lemma 4.1.1 of \\cite{k}, namely, a geometric one which does not use\nthe classification of simple Lie algebras over ${\\mathbb C}$. Indeed, as explained on\np.~14 of {\\em op.~cit.}, the lemma reduces to the statement that, with our\nnotation, if we choose the embedding $\\iota:{\\mathbb Q}\/{\\mathbb Z}\\hookrightarrow\\bC^\\times$ defined by\n$\\iota(q)=\\exp(-2\\pi\\sqrt{-1}\\cdot q)$ (this is the same choice as the one that\nwas made in Section \\ref{s:systems}, but is complex conjugate to the one used\nby Kottwitz in \\cite{k}), and the corresponding isomorphism $Z\\cong\nP(R^\\vee)\/Q(R^\\vee)$, then the representation of $Z$ on $\\operatorname{Lie}(A)^*$ obtained\nfrom the natural representation of $W$ via the homomorphism $Z\\longrightarrow W$,\n$x\\mapsto\\psi(\\mu_x)$, decomposes into the direct sum of the $1$-dimensional\nrepresentations given by the characters $\\varpi_i\\bigl\\lvert_Z$. However, basic\ncharacter theory of finite groups implies that it is enough to show that for\neach $x\\in Z$, the eigenvalues of $\\psi(\\mu_x)$ on $\\operatorname{Lie}(A)^*$ are the numbers\n$\\varpi_i(x)$, counting the multiplicities, and this is precisely the statement\ndual to part (c) of Theorem \\ref{t:s} above, in view of part (a) of Theorem\n\\ref{t:s}.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Geometric interpretation of $\\psi$}\\label{s:geometric}\n\n\nWe keep the conventions of \\S\\ref{ss:Conventions} and use the notation ($G$,\n$A$, $B$, $R$, $W$, etc.) introduced in \\S\\ref{ss:setup}. In this section we\nstudy in more detail the homomorphism $\\psi:P(R^\\vee)\/Q(R^\\vee)\\to W$\n(determined by the choice of $B$) and prove part (a) of Theorem \\ref{t:s}. The\nmain result of the section is Theorem \\ref{t:monodromy}, which provides a\ngeometric definition of $\\psi$.\n\n\n\\subsection{Classical definition of the homomorphism $\\psi$}\\label{ss:psi} We begin\nby recalling the definition of the homomorphism $\\psi$ that appears in\n\\cite{b}. Perhaps we provide a little more information than is strictly needed\nfor our purposes, but our presentation is somewhat clearer than Bourbaki's, and\nthe additional facts that we mention may help the reader understand the general\npicture.\n\n\\medbreak\n\nLet $V=Q(R)\\otimes_{\\mathbb Z}{\\mathbb R}$; note that our notation is different from the one used\nin \\cite{k}, since Kottwitz denotes by $V$ the complex space\n$Q(R)\\otimes_{\\mathbb Z}{\\mathbb C}$. With our notation, $R$ can be thought of as an abstract\nreduced and irreducible root system in the real vector space $V$ (\\cite{b},\nChapter VI, \\S1, no.~1). We introduce the corresponding affine root system\n$\\widetilde{R}$ in an ad hoc manner as follows. The root lattice of $\\widetilde{R}$ is by\ndefinition the lattice $Q(\\widetilde{R}):=Q(R)\\oplus{\\mathbb Z}\\cdot\\alpha_0$ in the vector space\n$\\widetilde{V}:=V\\oplus{\\mathbb R}\\cdot\\alpha_0$, where $\\alpha_0$ is just an auxiliary symbol. We call\n$\\alpha_0$ the {\\em affine simple root}. Let $\\widetilde{\\al}$ denote the highest root of\n$R$, and define $\\delta=\\alpha_0+\\widetilde{\\al}\\in Q(\\widetilde{R})$. We set\n\\[\n\\widetilde{R} = \\bigl\\{ n\\delta + \\alpha \\,\\lvert\\, \\alpha\\in R,\\ n\\in{\\mathbb Z} \\bigr\\} \\bigcup \\bigl\\{\nm\\delta \\,\\lvert\\, m\\in{\\mathbb Z}\\setminus\\{0\\}\\}.\n\\]\nThis is the {\\em affine root system} associated to $R$. The elements of $\\widetilde{R}$\nof the form $m\\delta$, $m\\in{\\mathbb Z}\\setminus\\{0\\}$, are called {\\em imaginary roots};\nall other elements are called {\\em real} ({\\em affine}) {\\em roots}. A real\nroot $n\\delta+\\alpha$ is called {\\em Dynkin} if $n=0$, and {\\em non-Dynkin}\notherwise. Thus $R$ is identified with the subset of $\\widetilde{R}$ consisting of Dynkin\nroots.\n\n\\medbreak\n\nThe elements $\\{\\alpha_i\\}_{i\\in I\\cup \\{0\\}}$ are the {\\em simple roots} of the\nsystem $\\widetilde{R}$. Thus $\\alpha_0$ is the unique non-Dynkin simple root. The set\n$I\\cup\\{0\\}$ is in natural bijection with the set of vertices of the extended\nDynkin diagram corresponding to the root system $R$. Moreover, let us write\n\\[\n\\widetilde{\\al} = \\sum_{i\\in I} \\delta_i \\alpha_i, \\qquad \\delta_i\\in{\\mathbb N},\n\\]\n(note that Bourbaki uses a different notation: $\\widetilde{\\al}=\\sum n_i\\alpha_i$), and set\n$\\delta_0=1$. Then we have (by construction)\n\\[\n\\delta = \\sum_{i\\in I\\cup\\{0\\}} \\delta_i \\alpha_i.\n\\]\n\n\\medbreak\n\nWe identify $V$ with a quotient of $\\widetilde{V}$, namely, $\\widetilde{V}\\bigl\/({\\mathbb R}\\cdot\\delta)\\cong\nV$.  Explicitly, this isomorphism is the inverse of the composition of the\nnatural inclusion $V=V\\oplus(0)\\hookrightarrow\\widetilde{V}$, followed by the projection. This\nisomorphism takes the image of $\\alpha_i\\in\\widetilde{V}$ in $\\widetilde{V}\\bigl\/({\\mathbb R}\\cdot\\delta)$ to\n$\\alpha_i\\in V$ for all $i\\in I$, and it takes the image of $\\alpha_0\\in\\widetilde{V}$ in\n$\\widetilde{V}\\bigl\/({\\mathbb R}\\cdot\\delta)$ to $-\\widetilde{\\al}\\in V$. Dually, we identify $V^*$ with a\nsubspace of $\\widetilde{V}^*$:\n\\[\nV^* \\cong \\{ f\\in \\widetilde{V}^* \\,\\bigl\\lvert\\, f(\\delta)=0 \\}.\n\\]\nIn particular,\n\\begin{equation}\\label{e:Rvee}\nP(R^\\vee) \\cong \\{ f\\in\\widetilde{V}^* \\,\\bigl\\lvert\\, f(R)\\subseteq{\\mathbb Z},\\ \\ f(\\delta)=0 \\}.\n\\end{equation}\nWe also define\n\\[\nE = \\{ f\\in \\widetilde{V}^* \\,\\bigl\\lvert\\, f(\\delta)=1 \\}.\n\\]\nThis is clearly an affine space for the vector space $V^*$. We now define a\nlinear action $\\mu\\mapsto t_\\mu$ of $P(R^\\vee)$ on $\\widetilde{V}^*$ as follows: if\n$\\mu\\in P(R^\\vee)$ and $f\\in\\widetilde{V}^*$, then $t_\\mu(f)=f+f(\\delta)\\cdot\\mu$. Note that\nthis action preserves $E$, thanks to \\eqref{e:Rvee}, and the induced action of\n$P(R^\\vee)$ on $E$ is simply by translations. We also define $\\mu\\mapsto\nt^*_\\mu$ to be the contragredient action of $P(R^\\vee)$ on $\\widetilde{V}$. On the other\nhand, we define an action of the Weyl group $W$ on $\\widetilde{V}$ by declaring that $W$\nacts trivially on $\\delta$ and acts by its usual action on $V$. Then we also\nobtain the contragredient action of $W$ on $\\widetilde{V}^*$, and it is easy to check\nthat the action of $W$ on $\\widetilde{V}$ (resp., on $\\widetilde{V}^*$) normalizes the action of\n$P(R^\\vee)$ on $\\widetilde{V}$ (resp., on $\\widetilde{V}^*$); in fact, e.g., if $w\\in W$ and\n$\\mu\\in P(R^\\vee)$, then $wt^*_\\mu w^{-1} = t^*_{w(\\mu)}$. This equation\nimplies that we have an action of the {\\em extended Weyl group}\n$W_e:=P(R^\\vee)\\rtimes W$ on $\\widetilde{V}$ and $\\widetilde{V}^*$, and, moreover, this action\npreserves $E$. Note also that $W_e$ contains as a subgroup the {\\em affine Weyl\ngroup} $W_a:=Q(R^\\vee)\\rtimes W$.\n\n\\medbreak\n\nThe last key ingredient is the following fact. For every real affine root\n$\\alpha\\in\\widetilde{R}$, we have the corresponding root hyperplane\n$\\operatorname{Ker}(\\alpha)\\subseteq\\widetilde{V}^*$, and the intersection $\\operatorname{Ker}(\\alpha)\\cap E$ is a\nhyperplane in $E$. The complement of the union of all such affine root\nhyperplanes in $E$ is a disjoint union of bounded connected open subsets of\n$E$, called the {\\em alcoves}. Moreover, it is clear that the action of $W_e$\npermutes the real affine roots, hence also permutes the alcoves. A fundamental\nresult (\\cite{b}, Chapter VI, \\S2, no.~1) is that the affine Weyl group $W_a$\nalready acts {\\em simply transitively} on the alcoves. This implies that if $C$\nis a fixed alcove and $\\Gamma_C$ is the group of automorphisms of $C$ in $W_e$,\nthen $\\Gamma_C$ projects isomorphically onto the quotient $W_e\/W_a\\cong\nP(R^\\vee)\/Q(R^\\vee)$. Furthermore, we do have a canonical choice for $C$: it is\nthe so-called {\\em fundamental alcove}, defined by\n\\[\nC = \\bigl\\{ v\\in E \\,\\,\\bigl\\lvert\\,\\, 0<v(\\alpha_i)<1 \\ \\forall\\,i\\in I\\cup\\{0\\}\n\\bigr\\}.\n\\]\nIf $\\xi:\\Gamma_C\\longrightarrow P(R^\\vee)\/Q(R^\\vee)$ is the corresponding isomorphism and\n$\\mathrm{pr}:W_e\\longrightarrow W$ is the natural projection, then the sought-after homomorphism\n$\\psi:P(R^\\vee)\/Q(R^\\vee) \\longrightarrow W$ is defined as the composition\n$\\psi=\\mathrm{pr}\\circ\\xi^{-1}$. Note that the construction of $\\psi$ is completely\ncanonical once the basis of simple roots $\\{\\alpha_i\\}_{i\\in I}$ has been chosen.\n\n\\medbreak\n\nWe now state the main property of the homomorphism $\\psi$ that will be used\nbelow. Let us denote by $\\{\\overline{\\omega}_i\\}_{i\\in I}$ the basis of $P(R^\\vee)$ dual to\nthe basis $\\{\\alpha_i\\}_{i\\in I}$ of $Q(R)$. For notational convenience, we set\n$\\overline{\\omega}_0=0$. Following Bourbaki (\\cite{b}, Chapter VI, \\S2, no.~2), we define a\nsubset $J\\subseteq I$ by\n\\[\nJ = \\{ i\\in I \\,\\bigl\\lvert\\, \\delta_i=1 \\}.\n\\]\nFor example, $J=I$ if $R$ is of type $A$, and $J\\subsetneq I$ otherwise.\n\\begin{lem}\\label{l:psi} The elements $\\{\\overline{\\omega}_i\\}_{i\\in J\\cup\\{0\\}}$ form a\ncomplete set of representatives for the quotient $P(R^\\vee)\/Q(R^\\vee)$.\nMoreover, for each $i\\in J\\cup\\{0\\}$, we have\n$\\xi^{-1}\\left(\\left[\\overline{\\omega}_i\\right]\\right)=t_{\\overline{\\omega}_i}\\circ\\psi(\\overline{\\omega}_i)$, where\n$\\left[\\overline{\\omega}_i\\right]$ denotes the image of $\\overline{\\omega}_i$ in $P(R^\\vee)\/Q(R^\\vee)$, and\n$\\psi(\\overline{\\omega}_i)$ is viewed as an element of $W_e$ via the obvious inclusion\n$W\\hookrightarrow W_e$.\n\\end{lem}\n\\begin{proof}\nSee \\cite{b}, Chapter VI, \\S2, no.~3, Proposition 6 and\nits Corollary.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Interpretation of $\\psi$ in terms of monodromy}\\label{ss:monodromy} The\nhomomorphism $\\psi$ described above does not appear to be very well understood,\nsince the purely algebraic definition of \\cite{b} is not very enlightening. The\nresults of this section and the previous one are a first step towards\nunderstanding the homomorphism $\\psi$ in more concrete terms. One drawback of\nTheorem \\ref{t:s} is that it only describes the conjugacy classes of each of\nthe elements $\\psi(\\mu)$ separately, rather than the conjugacy class of the\nwhole homomorphism $\\psi$. This drawback is corrected in Theorem\n\\ref{t:monodromy} below.\n\n\\medbreak\n\nWe now concentrate on the question of whether the homomorphism $\\psi$ admits a\ngeometric, rather than algebraic, description. One way to state this question\nprecisely is as follows: do there exist a smooth complex algebraic variety $Y$\nwhose fundamental group (with respect to the complex topology) is isomorphic to\n$P(R^\\vee)\/Q(R^\\vee)$ and a $W$-torsor $X$ over $Y$ such that the induced\nmonodromy homomorphism $P(R^\\vee)\/Q(R^\\vee)\\longrightarrow W$ is conjugate to $\\psi$? We\ndo not know a definite answer to this very question, but we do have a\n``semi-positive'' answer, which may be sufficient for many practical purposes.\nThe formulation of the answer is due to Ilya Zakharevich. The main idea is as\nfollows.\n\n\\medbreak\n\nLet $G_{ad}=G\/Z$ be the simple adjoint group corresponding to $G$, let\n$G^{rs}\\subset G$ and $G^{rs}_{ad}\\subset G_{ad}$ be the (Zariski open) subsets\nconsisting of the regular semisimple elements of the groups $G$ and $G_{ad}$,\nand consider the following diagram:\n\\begin{equation}\\label{e:coverings}\n\\xymatrix{\n  \\ar[dd]_{p'} \\widetilde{G_{ad}^{rs}}  &     & \\ar[ll]_{q'} \\ar[dd]^p \\widetilde{G^{rs}}   \\\\\n     &        &     \\\\\n   G_{ad}^{rs}   &     &   \\ar[ll]_q G^{rs}\n   }\n\\end{equation}\nHere $\\widetilde{G^{rs}}$ denotes the incidence variety of pairs $(g,B')$\nconsisting of an element $g\\in G^{rs}$ and a Borel subgroup $B'\\subset G$\ncontaining $g$, and $p$ is the obvious projection. It is well known that $p$ is\na $W$-torsor, and, in particular, topologically it is a covering map. The\n$W$-torsor $p':\\widetilde{G^{rs}_{ad}}\\longrightarrow G^{rs}_{ad}$ is defined in a similar\nmanner; $q$ is induced by the quotient map $G\\longrightarrow G_{ad}$, and $q'$ is induced\nby $q$ in the obvious way. All maps in the diagram above are surjective, and\n$q$ can also be thought of as a $Z$-torsor.\n\n\\medbreak\n\nThe monodromy of $p'$ yields a homomorphism $\\tau:\\pi_1(G_{ad}^{rs})\\longrightarrow W$,\nwhich is well defined up to conjugation. On the other hand, $q$ induces an\nexact sequence of groups\n\\begin{equation}\\label{e:fund-grps}\n\\pi_1(G^{rs}) \\rarab{q_*} \\pi_1(G^{rs}_{ad}) \\longrightarrow Z \\longrightarrow 1.\n\\end{equation}\nOf course, if the composition $\\tau\\circ q_*$ were trivial, then $\\tau$ would\ninduce a homomorphism $Z\\longrightarrow W$. However, this is far from being true, for the\ntriviality of $\\tau\\circ q_*$ would imply that $p$ is a trivial $W$-torsor,\nwhich is certainly false. Instead, we proceed as follows. Let us choose a\ncompact form $K\\subset G$ of $G$, and write $K_{ad}=K\/Z$, which is a compact\nform of $G_{ad}$. The diagram \\eqref{e:coverings} can be restricted to the\n``compact part'', which produces\n\\begin{equation}\\label{e:coverings-compact}\n\\xymatrix{\n  \\ar[dd]_{p'_K} \\widetilde{K_{ad}^{rs}}  &     & \\ar[ll]_{q'_K} \\ar[dd]^{p_K} \\widetilde{K^{rs}}   \\\\\n     &        &     \\\\\n   K_{ad}^{rs}   &     &   \\ar[ll]_{q_K} K^{rs}\n   }\n\\end{equation}\nOf course, $K^{rs}$ (resp., $K^{rs}_{ad}$) just stands for the set of regular\nelements of $K$ (resp., $K_{ad}$), since all elements of $K$ and $K_{ad}$ are\nautomatically semisimple. The advantage of considering this diagram lies in the following\n\\begin{lem}\\label{l:codim}\nThe complement $K\\setminus K^{rs}$ has real codimension $\\geq 3$ in $K$.\n\\end{lem}\n\\begin{proof}\nLet $T=A\\cap K$, which is a maximal torus of $K$. Let $T^{reg}=K^{rs}\\cap T$\ndenote the set of regular elements of $T$. It is well known that every element\n$y\\in K$ is conjugate to an element of $T$, and $y$ is regular if and only if\nit is conjugate to an element of $T^{reg}$. Thus the conjugation map\n\\[\nK \\times \\bigl(T\\setminus T^{reg}\\bigr) \\longrightarrow K\\setminus K^{rs}\n\\]\nis surjective. Moreover, its fibers have dimension $\\geq T+2$, because if $t\\in\nT$ is not regular, its centralizer in $K$ contains $T$ as well as a nontrivial\ncompact semisimple Lie group. Hence\n\\[\n\\operatorname{dim}(K\\setminus K^{rs}) \\leq \\operatorname{dim}(K) + \\operatorname{dim}(T\\setminus T^{reg}) - \\operatorname{dim}(T) -2 \\leq \\operatorname{dim}(K)-3,\n\\]\ncompleting the proof.\n\\end{proof}\n\nA standard fact in algebraic topology now yields\n\n\\begin{cor}\\label{c:fund}\nThe inclusions $K^{rs}\\hookrightarrow K$ and $K_{ad}^{rs}\\hookrightarrow K_{ad}$ induce isomorphisms on the fundamental groups.\n\\end{cor}\n\nThus $K^{rs}$ is simply connected, and the natural map $\\pi_1(K^{rs}_{ad})\\longrightarrow\nZ$ is an isomorphism. In particular, the monodromy of $p'_K$ does determine a\nconjugacy class of homomorphisms $\\tau_K : Z \\longrightarrow W$. Our last main result is\n\n\\begin{thm}\\label{t:monodromy}\nIn the situation of Theorem $\\ref{t:s}$, there exists a regular element $u\\in\nA\\cap K$ such that $\\psi(\\mu_x)(u)=x u$ for all $x\\in Z$. If $\\overline{u}$\ndenotes its image in $K_{ad}^{rs}$, then the corresponding monodromy\nhomomorphism $\\tau_K:Z\\cong\\pi_1(K_{ad}^{rs},\\overline{u})\\longrightarrow W$ is conjugate\nto the homomorphism $x\\mapsto\\psi(\\mu_x)$.\n\\end{thm}\n\nNote that this theorem is not quite what we were looking for, since the\nconstruction is based on the compact form $K$, and hence cannot be formulated\nin a completely algebro-geometric way. The situation can be improved slightly\nby noting that the inclusion $K^{rs}_{ad}\\hookrightarrow G^{rs}_{ad}$ induces a\nhomomorphism\n\\[\nj : Z\\rarab{\\simeq}\\pi_1(K^{rs}_{ad})\\longrightarrow\\pi_1(G^{rs}_{ad})\n\\]\nwhich is a splitting of \\eqref{e:fund-grps}. Hence the homomorphism\n$x\\mapsto\\psi(\\mu_x)$ is also conjugate to the composition of the monodromy of\nthe algebraic covering $p':\\widetilde{G^{rs}_{ad}}\\longrightarrow G^{rs}_{ad}$ with $j$.\nUnfortunately, the definition of $j$ still involves $K$; however, if we could\ncharacterize abstractly all splittings $j$ of \\eqref{e:fund-grps} that arise in\nthis way, then we would have a manageable algebro-geometric description of the\nhomomorphism $\\psi:P(R^\\vee)\/Q(R^\\vee)\\longrightarrow W$.\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{t:monodromy}}\\label{ss:proof-monodromy} The proof\nof the first statement of Theorem \\ref{t:monodromy} that we give below was\nexplained to us by R.~Kottwitz (it is much faster and more transparent than our\noriginal argument). A similar idea was suggested independently by J.-P.~Serre.\n\n\\medbreak\n\nTo simplify notation, we assume that the embedding $\\iota:{\\mathbb Q}\/{\\mathbb Z}\\hookrightarrow\\bC^\\times$ is\nchosen to be the same as the one used in Section \\ref{s:systems}, namely,\n$\\iota(q)=\\exp(-2\\pi\\sqrt{-1}\\cdot q)$. Let $T=A\\cap K$, as before. There is a\nnatural identification $T=\\operatorname{Hom}(P(R),S^1)$. If $V$, $\\widetilde{V}$ and\n$E\\subset \\widetilde{V}^*$ are as in \\S\\ref{ss:psi}, we define an ``exponential\nmap''\n\\[\n\\operatorname{Exp} : E \\longrightarrow T\n\\]\nby the formula\n\\[\n\\operatorname{Exp}(f)(v)=\\exp\\bigl(2\\pi\\sqrt{-1}\\cdot f(v)\\bigr) \\qquad \\forall\\, v\\in P(R);\n\\]\nit identifies $T$ with the quotient of $E$ by the translation action of\n$Q(R^\\vee)$. In particular, let $e\\in C$ denote the barycenter of the\nfundamental alcove $C\\subset E$ introduced in \\S\\ref{ss:psi} (i.e., $e$ is the\narithmetic mean of the vertices of $C$). Since $e$ does not lie in any of the\naffine root hyperplanes in $E$, the element $u:=\\operatorname{Exp}(e)\\in T$ is regular. On\nthe other hand, $\\Gamma_C$ fixes $e$ because it preserves $C$. According to Lemma\n\\ref{l:psi}, this means that $t_{-\\nu}(e)\\equiv\\psi(\\nu)(e)$ modulo $Q(R^\\vee)$\nfor all $\\nu\\in P(R^\\vee)\/Q(R^\\vee)$, which is equivalent to the property\n$xu=\\psi(\\mu_x)(u)$ for all $x\\in Z$. This proves the first statement of\nTheorem \\ref{t:monodromy}.\n\n\\medbreak\n\nWe now prove the second statement. Let $u$ be as above, and let $\\overline{u}$\ndenote the image of $u$ in $K^{rs}_{ad}$. For each $x\\in Z$, choose a\ncontinuous path $\\gamma_x$ from $u$ to $xu$ in $K^{rs}$, and let\n$\\overline{\\gamma}_x$ denote its image in $K^{rs}_{ad}$, which is a loop at\n$\\overline{u}$. Then the map $Z\\longrightarrow \\pi_1(K^{ad}_{rs},\\overline{u})$ given by\n$x\\mapsto[\\overline{\\gamma}_x]$ is a group isomorphism by Corollary \\ref{c:fund}.\nSince $K^{rs}$ is simply connected, the projection $p_K:\\widetilde{K}^{rs}\\to\nK^{rs}$ has a (unique) continuous section $\\sigma_K:K^{rs}\\to\\widetilde{K}^{rs}$\nsatisfying $\\sigma_K(u)=(u,B)$, where $B$ is our chosen Borel subgroup of $G$\ncontaining $A\\ni u$. The second statement of Theorem \\ref{t:monodromy} is\nequivalent to the assertion that $\\sigma_K(xu)=(xu,B^{\\psi(\\mu_x)})$ for all $x\\in\nZ$, where $B^{\\psi(\\mu_x)}$ is the Borel subgroup obtained by conjugating $B$\nwith the Weyl group element $\\psi(\\mu_x)\\in W$. However, $xu=\\psi(\\mu_x)(u)$ by\nconstruction, so the equality $\\sigma_K(xu)=(xu,B^{\\psi(\\mu_x)})$ follows from the\nstronger assertion that $\\sigma_K(gug^{-1})=(gug^{-1},B^g)$ for all $g\\in K$. To\nverify the last equality, observe that it holds for $g=1$ by assumption, and\ntherefore holds for all $g\\in K$ by connectedness. This proves Theorem\n\\ref{t:monodromy}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe consider the problem of uniformly tessellating sets in a high-dimensional Euclidean space using random hyperplanes. In what follows we use the following notation. For any $a\\in \\mathbb{R}^n$ and $\\tau\\in \\mathbb{R}$ let\n$$\nH(a,\\tau) = \\{x\\in \\mathbb{R}^n \\ : \\ \\langle a,x\\rangle+\\tau=0\\}\n$$\ndenote the hyperplane defined by $(a,\\tau)$. Each hyperplane $H(a,\\tau)$ divides $\\mathbb{R}^n$ into two parts---the positive and the negative ones, and as a result a collection of hyperplanes endows a tessellation of any given subset $T\\subset \\mathbb{R}^n$.\n \nLet $A:\\mathbb{R}^n\\to \\mathbb{R}^m$ be the matrix with rows $a_1^T,\\ldots,a_m^T$, let $\\tau=(\\tau_1,\\ldots,\\tau_m)\\in \\mathbb{R}^m$, and let $\\operatorname{sign}:\\mathbb{R}^m\\to \\{-1,1\\}^m$ be defined by $\\operatorname{sign}(y)=(\\operatorname{sign}(y_i))_{i=1}^m$. In this notation, for any given $x\\in T$, the bit string $\\operatorname{sign}(Ax+\\tau)\\in \\{-1,1\\}^m$ encodes the ``cell\" in which $x$ is located. Moreover, the normalized Hamming distance\n$$\n\\frac{1}{m}d_H(\\operatorname{sign}(Ax+\\tau),\\operatorname{sign}(Ay+\\tau))\n$$\nis the fraction of the $m$ hyperplanes that separate $x$ and $y$.\n\\begin{Definition} \\label{def:tess}\nThe hyperplanes $H(a_i,\\tau_i)$, $i=1,\\ldots,m$, endow a $\\delta$-tessellation of the set $T$ if, for some constant $\\kappa$,\n\\begin{equation}\n\\label{eqn:deltaTessProp}\n\\left|\\frac{\\kappa}{m}d_H(\\operatorname{sign}(Ax+\\tau),\\operatorname{sign}(Ay+\\tau)) - \\|x-y\\|_2\\right|\\leq \\delta, \\qquad \\text{for every} \\  x,y \\in T.\n\\end{equation}\n\\vspace{0.1cm}\n\\end{Definition}\n\nIn other words, a $\\delta$-tessellation implies that the Euclidean distance between any two points is proportional to the fraction of hyperplanes separating the two, up to a small additive error of $\\delta$. In particular, the Euclidean diameter of all the cells of a $\\delta$-tessellation of $T$ is at most $\\delta$. Phrased differently, such a $\\delta$-tessellation means that the map $f:\\mathbb{R}^n\\to \\{-1,1\\}^m$, $f(t)=\\operatorname{sign}(At+\\tau)$, is a \\emph{$\\delta$-embedding}; an almost-isometric embedding of $T$ into the discrete cube, with the latter endowed with the metric $\\frac{\\kappa}{m}d_H$. It is important to stress that an `almost-isometric' embedding here means isometric up to an \\emph{additive error}, rather that a multiplicative one.\n\\par\nApart from being of intrinsic geometric interest, random hyperplanes tessellations and their associated random embeddings appear naturally in at least two applications. First, a $\\delta$-embedding can clearly be used to compress a dataset $T$ into a small number of bits. Property \\eqref{eqn:deltaTessProp} ensures that one can quickly retrieve information on Euclidean distances between the original data up to a small error---an important feature for many computational tasks. In this application, the number of hyperplanes $m$ determines the degree of compression that is achieved. Second, the map $t\\mapsto\\operatorname{sign}(At+\\tau)$ occurs as a simple and popular model in signal processing. In this context, $Ax$ models the analog measurements of a signal $x$ and the map $y\\mapsto \\operatorname{sign}(y+\\tau)$ a coarse quantizer that converts $Ax$ into a digital signal. Property \\eqref{eqn:deltaTessProp} plays an essential role in proving error bounds for algorithms that aim to reconstruct a signal $x$ from its quantized measurements $\\operatorname{sign}(Ax+\\tau)$. In this second application, $m$ corresponds to the number of measurements of the signal. In both applications, it is of significant interest to understand which $m$ is necessary and sufficient to guarantee that $f$ is a $\\delta$-embedding of a given set $T$. We refer to \\cite{DM18,PlV14} and the references therein for a more extensive discussion of these applications.\\par   \nIn this work we focus on what is arguably the most natural random hyperplane tessellation---with directions selected according to the standard gaussian distribution. Motivated by the aforementioned applications, our goal is to derive sharp bounds on the number of such hyperplanes that are needed to generate a $\\delta$-tessellation. This problem was first studied by Plan and Vershynin \\cite{PlV14} for subsets of the Euclidean sphere $S^{n-1}$, and with hyperplanes that pass through the origin (i.e., $\\tau_i=0$ for $i=1,\\ldots,m$).\n\\begin{Remark}\nIt is important to stress that hyperplanes that pass through the origin cannot separate points on any ray $\\{ \\alpha x : \\alpha \\geq 0\\}$. As a result, shifts must be used in a $\\delta$-tessellation of an arbitrary set.\n\\end{Remark}\n\n\nTo formulate the result from \\cite{PlV14}, denote by $G$ the standard gaussian random vector and set\n$$\n\\ell_*(T):= \\mathbb{E} \\sup_{t \\in T} |\\inr{G,t}| \n$$\ndenote the \\emph{gaussian mean width} of a set $T\\subset \\mathbb{R}^n$. Also, let\n$$\nd_{S^{n-1}}(x,y)=\\frac{1}{\\pi}\\arccos\\left(\\frac{\\inr{x,y}}{\\|x\\|_2\\|y\\|_2}\\right)\n$$\nbe the spherical distance.\n\n\\begin{Theorem} \\label{thm:PV} \\cite{PlV14}\nThere are absolute constants $c_1$ and $c_2$ such that the following holds. Let $T \\subset S^{n-1}$ and let $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ be the gaussian matrix. For $\\delta>0$, if\n$$\nm \\geq c_1 \\frac{\\ell_*^2(T)}{\\delta^6}\n$$\nthen with probability at least $1-2e^{-c_2m\\delta^2}$, for every $x,y\\in T$,\n\\begin{equation}\n\\label{eqn:PlVEmbed}\n\\left|\\frac{1}{m}d_H(\\operatorname{sign}(Ax),\\operatorname{sign}(Ay)) - d_{S^{n-1}}(x,y)\\right|\\leq \\delta.\n\\end{equation}\n\\end{Theorem}\n\\begin{Remark}\nIt was conjectured by Plan and Vershynin that the optimal bound on $m$ should be $m\\sim \\ell_*^2(T)\/\\delta^2$.\n\\end{Remark}\nAn improvement on the estimate from Theorem \\ref{thm:PV} can be found in \\cite{OyR15} (see Theorem~2.5 there). To formulate it, let ${\\cal N} (T,\\varepsilon)$ be the covering number at scale $\\varepsilon$ with respect to the Euclidean norm; that is, the smallest number of open Euclidean balls of radius $\\varepsilon$ needed to cover $T$. Moreover, set $T-T=\\{t_1-t_2 : t_1,t_2 \\in T\\}$.\n\\begin{Theorem} \\cite{OyR15}\n\\label{thm:OyR}\nThere are absolute constants $c_1,c_2,$ and $c_3$ such that the following holds. Let $T \\subset S^{n-1}$ and let $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ be the gaussian matrix. For $\\delta>0$ let\n$$\n0<\\theta \\leq c_1 \\frac{\\delta}{\\sqrt{\\log(e\/\\delta)}}\n$$\nand set\n\\begin{equation} \\label{eq:est-OyR15}\nm \\geq c_2 \\left(\\frac{\\log{\\cal N}(T,\\theta)}{\\delta^2} + \\frac{\\ell_*^2((T-T)\\cap \\theta B_2^n)}{\\delta^3}\\right).\n\\end{equation}\nThen with probability at least $1-2e^{-c_3 m\\delta^2}$, for every $x,y\\in T$,\n$$\n\\left|\\frac{1}{m}d_H(\\operatorname{sign}(Ax),\\operatorname{sign}(Ay)) - d_{S^{n-1}}(x,y)\\right|\\leq \\delta.\n$$\n\\end{Theorem}\n\nAs noted previously, the hyperplanes in Theorems~\\ref{thm:PV} and \\ref{thm:OyR} cannot be used to generate a $\\delta$-tessellation of an arbitrary set  $\\mathbb{R}^n$. A natural way of addressing that issue is to consider parallel shifts---meaning that $\\tau_1,\\ldots,\\tau_m$ need not be $0$. As it happens, under relatively mild randomness assumptions on $A$, \\emph{random shifts} prove to be effective---leading to hyperplane tessellations that endow multiplicative \\emph{isomorphic} embeddings in $\\{-1,1\\}^m$.\n\nTo formulate that fact, recall that a random vector $X$ is isotropic if for every $t \\in \\mathbb{R}^n$, $\\mathbb{E} \\inr{X,t}^2 = \\|t\\|_2^2$. The random vector is $L$-subgaussian if for every $p \\geq 2$ and $t\\in \\mathbb{R}^n$,\n$$\n(\\mathbb{E} |\\langle X,t\\rangle|^p )^{1\/p} \\leq L \\sqrt{p} (\\mathbb{E} \\langle X,t\\rangle^2)^{1\/2}.\n$$\nLet $A$ be a matrix whose rows are independent copies of a symmetric, isotropic, $L$-subgaussian random vector and let $\\tau_i$ be independent random variables, which are distributed uniformly in $[-\\lambda,\\lambda]$ and are independent of $A$.\n\n\\begin{Theorem} \\label{thm:DM} \\cite{DM18}\nThere are constants $c_0,...,c_4$ that depend only on $L$ such that the following holds. Let $T \\subset \\mathbb{R}^n$ and set $R=\\sup_{t \\in T} \\|t\\|_2$. Put $\\lambda = c_0 R$ and set\n\\begin{equation} \\label{eqn:bitCompDM18}\nm \\geq c_1R \\log(eR\/\\delta) \\frac{\\ell_*^2(T)}{\\delta^3}.\n\\end{equation}\nThen with probability at least $1-8\\exp(-c_2m\\delta\/R)$, for any $x,y \\in {\\rm conv}(T)$ that satisfy $\\|x-y\\|_2 \\geq \\delta$, one has\n\\begin{equation} \\label{eq:isomorphic}\nc_3\\frac{ \\|x-y\\|_2}{R} \\leq \\frac{1}{m}d_H(\\operatorname{sign}(Ax+\\tau),\\operatorname{sign}(Ay+\\tau)) \\leq c_4\\sqrt{\\log(eR\/\\delta)} \\cdot \\frac{\\|x-y\\|_2}{R}.\n\\end{equation}\n\\end{Theorem}\n\nIt is unrealistic to hope for an almost isometric estimate in the context of Theorem \\ref{thm:DM} when using an arbitrary subgaussian matrix $A$ (because of the behaviour of the means $\\mathbb{E} d_H(\\operatorname{sign}(Ax+\\tau),\\operatorname{sign}(Ay+\\tau))$. But as it happens, our first result is that if $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ is a gaussian matrix and $m$ is well-chosen, then the random mapping $t \\mapsto \\operatorname{sign}(At+\\tau)$ is an almost isometric embedding (in the additive error sense) of an arbitrary $T \\subset \\mathbb{R}^n$. Moreover, the estimate on $m$ is of a similar nature to the one from \\eqref{eq:est-OyR15}.\n\n\\begin{Theorem} \\label{thm:main-A}\nThere exist absolute constants $c_0,...,c_3$ such that the following holds. Let $T \\subset \\mathbb{R}^n$ and put $R=\\sup_{t \\in T} \\|t\\|_2$. Set $\\delta \\in (0,\\frac{R}{2}]$, $u\\geq 1$ and let\n$$\n0<\\theta \\leq c_0 \\frac{\\delta}{\\sqrt{\\log(e\\lambda\/\\delta)}}.\n$$\nConsider $\\lambda \\geq c_1 R\\sqrt{\\log(R\/\\delta)}$ and\n\\begin{equation}\n\\label{eqn:main-ABdm}\nm\\geq c_2 \\left( \\lambda^2 \\frac{\\log {\\cal N}(T,\\theta)}{\\delta^2}  +  \\lambda \\frac{\\ell_*^2((T-T)\\cap \\theta B^n_2)}{\\delta^3} \\right).\n\\end{equation}\nIf $A:\\mathbb{R}^m \\to \\mathbb{R}^n$ is the standard gaussian matrix and $\\tau$ is uniformly distributed in $[-\\lambda,\\lambda]^m$, then with probability at least $1-2\\exp(-c_3\\delta^2m\/\\lambda^2)$, the map $f(t)= \\operatorname{sign}(At+\\tau)$ satisfies\n\\begin{equation} \\label{eqn:GaussianDitheredText}\n\\sup_{x,y\\in T}\\left|\\frac{\\sqrt{2\\pi}\\lambda}{m}d_H(f(x),f(y))-\\|x-y\\|_2\\right|\\leq  \\delta.\n\\end{equation}\n\\end{Theorem}\n\nThe proof of Theorem \\ref{thm:main-A} follows from a generic embedding result, presented in Section~\\ref{sec:generic}. We show that if a deterministic matrix $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ `acts well' on $T$ (in a sense that is clarified in what follows) and $\\tau$ is as in Theorem \\ref{thm:main-A}, then with high probability with respect to $\\tau$, the mapping $t \\mapsto \\operatorname{sign}(At+\\tau)$ is a $\\delta$-embedding of $T$. We then show in Section \\ref{sec:gaussian} that for any $T \\subset \\mathbb{R}^n$, a typical realization of the gaussian matrix $A$ `acts well' on $T$.\n\n\\vskip0.3cm\n\nThe choice of $\\lambda$ is very natural --- random shifts at a scale proportional to the radius of the set ensure that one can separate points that belong to the same ray $\\{\\alpha x : \\alpha \\geq 0\\}$. In fact, $x \\mapsto \\operatorname{sign}(Ax+\\tau)$ can already fail to be a $\\delta$-embedding with high probability on sets consisting of only two points on a ray if $\\lambda$ is smaller than the radius (see Appendix~\\ref{sec:minShift} for details). At the same time, the estimate \\eqref{eqn:main-ABdm} in Theorem \\ref{thm:main-A} looks anything but natural. In fact, it is reasonable to guess that it is suboptimal. It was conjectured that the right tradeoff between the accuracy parameter $\\delta$, the set $T$ and the dimension $m$ is $m \\sim \\ell_*^2(T)\/\\delta^2$---just as in the conjecture regarding subsets of the sphere.\n\\begin{tcolorbox}\nSurprisingly, the assertion of Theorem \\ref{thm:main-A} is optimal (up to logarithmic factors). And with the natural choice of $\\lambda$, the conjectured optimal estimate of $m \\sim \\ell_*^2(T)\/\\delta^2$ is simply false.\n\\end{tcolorbox}\nWe prove that fact by establishing two lower bounds, the first of which deals with properties of arbitrary random embeddings into the discrete cube.\n\n\\begin{Definition} \\label{def:random-emb}\nLet $(f,\\phi)$ be a pair of maps, where $f:B_2^n \\to \\{-1,1\\}^m$ and $\\phi:\\{-1,1\\}^m \\to \\mathbb{R}$. $(f,\\phi)$ is a \\emph{$\\delta$-inner product-preserving embedding} of $W \\subset B_2^n$ if \n\\begin{equation}\n\\label{eqn:defInnProdPres}\n\\sup_{(x,y) \\in W \\times W} \\left| \\phi(f(x),f(y)) - \\langle x,y\\rangle \\right| < \\delta.\n\\end{equation}\nA pair of random maps $(f,\\phi)$ is a \\emph{random inner product-preserving embedding into $\\{-1,1\\}^m$ with parameters $n,\\delta,\\eta,$ and $N$} if, for any $W \\subset B_2^n$ of cardinality $N$, \\eqref{eqn:defInnProdPres} holds with probability at least $1-\\eta$.\n\\end{Definition}\nIn what follows we abuse notation by omitting the dependence of the embedding on the parameters $n$, $\\eta$ and $N$ and keeping track only of the accuracy parameter $\\delta$ and the embedding dimension $m$. Using a polarization argument, it is straightforward to verify that a $\\delta$-embedding of $W\\cup(-W)$ yields a $\\delta$-inner product-preserving embedding of $W$. In particular, the map in Theorem~\\ref{thm:main-A} yields a random inner product-preserving embedding in the sense of Definition \\ref{def:random-emb} if\n\\begin{equation} \\label{eq:est-m-rand-emb}\nm\\sim \\log(e\/\\delta)\\frac{\\log(eN\/\\eta)}{\\delta^2}.\n\\end{equation}\nThe first lower bound we present shows that \\eqref{eq:est-m-rand-emb} is not improvable (up to a logarithmic factor).\n\\begin{Theorem} \\label{thm:main-B-1}\nThere are absolute constants $c_0,c_1,c_2,c_3$ such that the following holds. Let $N$ be an integer and set $0<\\eta<1$. Let $c_0(\\eta\/N)^{1\/2} \\leq \\delta \\leq c_1$ and put $n=\\frac{c_2}{\\delta^2}\\log(eN\/\\eta)$. If $(f,\\phi)$ is a random inner product-preserving embedding into $\\{-1,1\\}^m$ with parameters $n,\\delta,\\eta,N$ then\n$$\nm \\geq c_3 \\frac{\\log (eN\/\\eta)}{\\delta^2}.\n$$\n\\end{Theorem}\n\nThe proof of Theorem \\ref{thm:main-B-1} actually reveals more than is stated: for any random embedding as above, there is a `bad set' $W$ that forces the embedding dimension to be at least $\\delta^{-2} \\log (N\/\\eta)$. Moreover, there is an absolute constant $\\alpha$ so that $\\hat{W}=W\\cup(-W)$ is $\\alpha$-separated. In particular, if $\\delta$ is sufficiently small, then $(\\hat{W}-\\hat{W}) \\cap \\delta B_2^n = \\{0\\}$. As a result, for this bad set the dominating term in the upper estimate on the embedding dimension in Theorem \\ref{thm:main-A} is the entropic one, showing that it cannot be improved (up to logarithmic factors)---regardless of the choice of the random embedding.\n\nThe proof, which can be found in Section \\ref{sec:proof-B1}, uses similar ideas to the ones appearing in a construction due to Alon and Klartag \\cite{AlK17}. Alon and Klartag studied $\\delta$-distance sketches, which are data structures that allow the reconstruction of all Euclidean scalar products between points in a subset of $\\mathbb{R}^n$ of a given cardinality.\n\n\n\\vskip0.3cm\n\n\nIn the second lower bound, our focus is on the specific embedding $t \\mapsto \\operatorname{sign}(At+\\tau)$ that is used in Theorem \\ref{thm:main-A}. We show that for any convex body\\footnote{A convex body in $\\mathbb{R}^n$ is a convex, centrally-symmetric set with a nonempty interior.} $T \\subset \\mathbb{R}^n$, the best possible dimension $m$ one can hope for coincides with the estimate from Theorem \\ref{thm:main-A} (up to logarithmic factors). To formulate that lower bound, set\n$$\nd^*(T) = \\left(\\frac{\\ell_*(T)}{\\sup_{t \\in T}\\|t\\|_2}\\right)^2\n$$\nto be the \\emph{Dvoretzky-Milman dimension} of $T$.\n\\begin{Theorem} \\label{thm:main-B-2}\nThere are absolute constants $c_0$, $c_1$ and $c_2$ such that the following holds. For any convex body $T \\subset \\mathbb{R}^n$, if $\\lambda \\geq c_0 \\ell_*((T-T)\\cap \\delta B^n_2)\/\\sqrt{m}$,\n$$\nm \\leq c_1 \\lambda \\frac{\\ell_*^2((T-T)\\cap \\delta B^n_2)}{\\delta^3},\n$$\nand $f$ is as in Theorem~\\ref{thm:main-A}, then with probability at least $1-2\\exp(-c_2d^*(T))$ there are $x,y \\in T$ such that\n$$\n\\left|\\frac{\\sqrt{2\\pi}\\lambda}{m}d_H(f(x),f(y))-\\|x-y\\|_2\\right| \\geq 2\\delta.\n$$\n\\end{Theorem}\n\n\\begin{Remark}\nThe lower bound on $\\lambda$ in Theorem~\\ref{thm:main-B-2} is not really restrictive. As an example, we will present in Lemma~\\ref{lemma:lower-on-lambda} a rather general situation in which that lower bound holds. \n\\end{Remark}\n\n\nThere is a purely geometric reason for Theorem \\ref{thm:main-B-2} being true. We first show that it is hard to embed a Euclidean ball using the mapping $y \\mapsto \\operatorname{sign}(y + \\tau)$. More accurately, if there is a subset of coordinates $I \\subset \\{1,...,m\\}$ such that $P_I V=\\{ (x_i)_{i \\in I} : x \\in V\\} =r B_2^I$, then embedding $V$ in $\\{-1,1\\}^m$ by using the mapping $y \\mapsto \\operatorname{sign}(y+\\tau)$ fails unless $m$ is sufficiently large (depending on $|I|$, $r$ and $\\lambda$). We then invoke the classical Dvoretzky-Milman Theorem and show that if $T \\subset \\mathbb{R}^n$ is a convex body and $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ is the standard gaussian matrix, then for a well-chosen $I$, with high probability, $P_I V=P_I A (T \\cap \\delta B_2^n)$ contains a large Euclidean ball. In particular, with high probability, the map $t \\mapsto \\operatorname{sign}(At + \\tau)$ fails to be a $\\delta$-embedding of $T$ unless $m$ is as in Theorem \\ref{thm:main-B-2}.\n\n\nThe proof of Theorem \\ref{thm:main-B-2} can be found in Section \\ref{sec:proof-B2}. The rest of that section is devoted to the conjecture we mentioned previously: that $m \\sim \\ell_*^2(T)\/\\delta^2$ suffices to ensure that $t \\mapsto \\operatorname{sign}(At + \\tau)$ is a $\\delta$-embedding of an arbitrary $T \\subset \\mathbb{R}^n$. Thanks to Theorem~\\ref{thm:main-B-2}, we construct an example which shows that the conjecture is false: a set for which $m \\sim \\ell_*^2(T)\/\\delta^3$ is required---see Theorem~\\ref{thm:counter-example}. \n\n\\section{A generic embedding result} \\label{sec:generic}\nThe first step in the proof of Theorem \\ref{thm:main-A} is actually rather general. We consider a set $T \\subset \\mathbb{R}^n$, and a (deterministic) matrix $A:\\mathbb{R}^n \\to \\mathbb{R}^m$. For a parameter $\\lambda>0$, let $\\tau$ be a random vector distributed uniformly  in $[-\\lambda,\\lambda]^m$, and the embedding function we consider is $f: \\mathbb{R}^n \\to \\{-1,1\\}^m$, defined by\n\\begin{equation} \\label{eqn:GaussBinEmdDef}\n f(x)=\\operatorname{sign} (Ax+\\tau).\n\\end{equation}\nHere, as always, the sign-function is applied component-wise. \n\nOur goal is to show that if $A$ acts on $T$ `in a regular way', then $f$ is a $\\delta$-embedding of $T$ into $\\{-1,1\\}^m$. To explain what we mean by `regularity', let $0 < \\theta < \\delta$, and set $T_\\theta \\subset T$ to be a minimal $\\theta$-net of $T$; in particular, $|T_\\theta| = {\\cal N}(T,\\theta)$. For $v \\in \\mathbb{R}^m$ and $1 \\leq s \\leq m$ set\n$$\n\\|v\\|_{[s]} = \\max_{|I|=s} \\left(\\sum_{i \\in I} v_i^2 \\right)^{1\/2},\n$$\nand assume that the following holds:\n\\begin{description}\n\\item{$(a)$} \\underline{uniform $\\ell_1$-concentration:} there is a constant $\\kappa$ such that\n\\begin{equation} \\label{eqn:ell1ell2assump}\n\\sup_{x,y\\in T_{\\theta}}\\left|\\frac{\\kappa}{m} \\|A(x-y)\\|_1 - \\|x-y\\|_2\\right|\\leq \\delta.\n\\end{equation}\n\\item{$(b)$} \\underline{$A$ maps $T$ to `well-spread' vectors:} For $s=\\lfloor\\delta m\/\\lambda\\rfloor$,\n\\begin{equation} \\label{eqn:knormbias}\n\\frac{1}{\\sqrt{s}} \\sup_{x\\in T_\\theta} \\|Ax\\|_{[s]}\\leq \\lambda,\n\\end{equation}\nand\n\\begin{equation} \\label{eqn:knormoscillations}\n\\frac{1}{\\sqrt{s}} \\sup_{x\\in (T-T)\\cap \\theta B_2^n} \\|Ax\\|_{[s]}\\leq \\delta.\n\\end{equation}\n\\end{description}\n\nDenote the normalized Hamming distance on $\\{-1,1\\}^m$ by\n\\begin{equation*}\n\\tilde{d}(x,y)= \\frac{2\\lambda \\kappa}{m}  d_H(x,y)\n\\end{equation*}\nfor a constant $\\kappa$ whose value is specified in what follows. In our application, where $A$ is the gaussian matrix, $\\kappa$ turns out to be an absolute constant. \n\\vskip0.3cm\nOnce $A$ exhibits the necessary regular behaviour, one may show that with high probability with respect to $\\tau$, $f$ is a $\\delta$-embedding of $T$.\n\n\\begin{Theorem} \\label{thm:mainGeneric}\nThere exist absolute constants $c_1,c_2,$ and $c_3$ such that the following holds. Let $\\theta \\leq \\delta$, set\n$$\nm \\geq c_1 \\lambda^2 \\kappa^2\\frac{\\log{\\cal N}(T,\\theta)}{\\delta^2}\n$$\nand assume that $A$ satisfies \\eqref{eqn:ell1ell2assump}, \\eqref{eqn:knormbias}, and \\eqref{eqn:knormoscillations}. Then with probability at least\n$$\n1-2 \\exp(-c_2 \\delta^2 m\/(\\lambda^2 \\kappa^2))\n$$\nwith respect to $\\tau$,\n\\begin{equation*}\n\\sup_{x,y\\in T}\\left|\\tilde{d}(f(x),f(y))-\\|x-y\\|_2\\right|\\leq c_3 (\\kappa+1) \\delta.\n\\end{equation*}\n\\end{Theorem}\n\n\nThe proof of Theorem \\ref{thm:mainGeneric} requires two simple observations.\n\\begin{Lemma}\\label{lem:exp}\nFix $\\lambda>0$ and let $\\tau$ be uniformly distributed in $[-\\lambda, \\lambda]$. If $\\phi_\\lambda(x)=(|x|-\\lambda)\\mathbbm{1}_{\\{|x|\\geq \\lambda\\}}$ then for every $a,b\\in \\mathbb{R}$,\n\\begin{equation*}\n\\left|2\\lambda \\mathbb{P}(\\operatorname{sign}(a+\\tau) \\neq \\operatorname{sign}(b+\\tau))- |a-b|\\right|\\leq \\phi_\\lambda(a)+\\phi_\\lambda(b).\n\\end{equation*}\n\\end{Lemma}\n\nThe proof of Lemma \\ref{lem:exp} follows from a straightforward and tedious computation. It is deferred to Appendix \\ref{app:proof-lem:exp}.\n\n\\begin{Corollary}  \\label{cor:exp}\nLet $x,y \\in T_\\theta$. If \\eqref{eqn:knormbias} holds then\n\\begin{equation*}\n\\left| \\mathbb{E}_\\tau d_H (f(x),f(y)) - \\frac{1}{2\\lambda} \\|A(x-y)\\|_1 \\right| \\leq s.\n\\end{equation*}\n\\end{Corollary}\n\n\n\\noindent {\\bf Proof.}\\ \\  Observe that\n\\begin{align*}\n\\mathbb{E}_\\tau d_H (f(x),f(y)) & =\\mathbb{E}_\\tau \\sum_{i=1}^m \\mathbbm{1}_{\\{ \\operatorname{sign}( (Ax)_i + \\tau_i) \\not= \\operatorname{sign}( (Ay)_i + \\tau_i)\\}}\n\\\\\n& = \\sum_{i=1}^m \\mathbb{P}_\\tau \\left(\\operatorname{sign}( (Ax)_i + \\tau_i) \\not= \\operatorname{sign}( (Ay)_i + \\tau_i)\\right),\n\\end{align*}\nand therefore, by Lemma \\ref{lem:exp},\n$$\n\\left| \\mathbb{E}_\\tau d_H (f(x),f(y)) - \\frac{1}{2\\lambda} \\|A(x-y)\\|_1 \\right|  \\leq \\frac{1}{\\lambda} \\sup_{z \\in T_\\theta}   \\sum_{i=1}^m |(Az)_i| \\mathbbm{1}_{\\{|(Az)_i|>\\lambda\\}}.\n$$\nTo control the right-hand side, note that by \\eqref{eqn:knormbias}, for every $z \\in T_\\theta$,\n\\begin{equation*} \n|(Az)^*_{s}| \\leq \\frac{1}{\\sqrt{s}} \\|Az\\|_{[s]}\\leq \\lambda,\n\\end{equation*}\nwhere $(Az)^*_{s}$ denotes the $s$-largest coordinate of $( |(Az)_i|)_{i=1}^m$. In particular,\n\\begin{equation} \\label{eqn:largeCoeffk-normBias}\n\\sup_{z\\in T_{\\theta}} \\left| \\left\\{i \\in \\{1,...,m\\} \\ : \\ |(Az)_i|> \\lambda \\right\\} \\right| \\leq s.\n\\end{equation}\nThus, using \\eqref{eqn:knormbias} once again,\n$$\n\\frac{1}{\\lambda}\\sup_{z \\in T_\\theta} \\sum_{i=1}^m |(Az)_i| \\mathbbm{1}_{\\{|(Az)_i|>\\lambda\\}} \\leq \\frac{1}{\\lambda}\\sup_{z \\in T_\\theta} \\sqrt{s} \\|Az\\|_{[s]} \\leq s.\n$$\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\n\\noindent{\\bf Proof of Theorem \\ref{thm:mainGeneric}.} For every $x\\in T$ let $\\pi x  \\in T_\\theta$ satisfy that $\\|x-\\pi x\\|_2 \\leq \\theta$. By the triangle inequality,\n\\begin{align}\\label{eq:four_summands}\n& \\left|\\tilde{d}(f(x),f(y))-\\|x-y\\|_2\\right| \\nonumber\\\\\n& \\qquad \\leq   \\left|\\tilde{d}(f(x),f(y))- \\tilde{d}(f(\\pi x),f(\\pi y))\\right| \\nonumber\n\\\\\n& \\qquad \\qquad + \\left|\\tilde{d}(f(\\pi x),f(\\pi y)) - \\mathbb{E}_{\\tau} \\tilde{d}(f(\\pi x),f(\\pi y)) \\right|\n+ \\left|\\mathbb{E}_{\\tau} \\tilde{d}(f(\\pi x),f(\\pi y)) - \\frac{\\kappa}{m}\\|A(\\pi x - \\pi y)\\|_1\\right| \\nonumber\n\\\\\n& \\qquad \\qquad + \\left|\\frac{\\kappa}{m}\\|A(\\pi x - \\pi y)\\|_1 - \\|\\pi x-\\pi y\\|_2\\right|\n+ \\left| \\|\\pi x-\\pi y\\|_2-\\|x-y\\|_2\\right| \\nonumber\n\\\\\n& \\qquad = (a)+(b)+(c)+(d)+(e).\n\\end{align}\nClearly, $(e)\\leq 2\\theta \\leq 2\\delta$ and, by \\eqref{eqn:ell1ell2assump}, $(d)\\leq \\delta$. Moreover, it follows from Corollary \\ref{cor:exp} that\n\\begin{align*}\n(c)= & \\left|\\mathbb{E}_{\\tau} \\tilde{d}(f(\\pi x),f(\\pi y)) - \\frac{\\kappa}{m}\\|A(\\pi x - \\pi y)\\|_1\\right|\n\\\\\n= & \\frac{2\\lambda \\kappa}{m} \\left|\\mathbb{E}_{\\tau} d_H(f(\\pi x),f(\\pi y)) - \\frac{1}{2\\lambda}\\|A(\\pi x - \\pi y)\\|_1\\right| \\leq 2\\lambda \\kappa \\frac{s}{m} \\leq 2 \\kappa \\delta.\n\\end{align*}\n\nTurning to $(b)$, note that\n$$\n\\tilde{d}(f(x),f(y))=\\frac{2\\lambda \\kappa}{m} \\sum_{i=1}^m \\mathbbm{1}_{\\{\\operatorname{sign}(Ax+\\tau)_i \\not = \\operatorname{sign}(Ay+\\tau)_i\\}},\n$$\nand in particular it is the sum of independent, $\\{0,1\\}$-valued random variables. Hence, by Hoeffding's inequality there is an absolute constant $c$ such that,  for every $x,y\\in T_\\theta$ and $\\delta>0$,\n\\begin{equation*}\n\\mathbb{P}_{\\tau}\\left(\\left|\\tilde{d}(f(\\pi x),f(\\pi y)) - \\mathbb{E}_{\\tau} \\tilde{d}(f(\\pi x),f(\\pi y))\\right| \\geq \\delta \\right)\\leq 2 \\exp\\left(-c\\frac{\\delta^2 m}{\\lambda^2 \\kappa^2}\\right).\n\\end{equation*}\nIt follows that if\n$$\nm \\geq c_1 \\kappa^2 \\frac{\\lambda^2}{\\delta^{2}} \\log(|T_{\\theta}|),\n$$\nthen by the union bound, with probability at least $1-2 \\exp(-c_2\\delta^2 m\/(\\lambda^2\\kappa^2))$,\n\\begin{equation*}\n\\sup_{x,y\\in T} \\left|\\tilde{d}(f(\\pi x),f(\\pi y)) - \\mathbb{E}_{\\tau} \\tilde{d}(f(\\pi x),f(\\pi y))\\right|\\leq \\delta.\n\\end{equation*}\nFinally, to control $(a)$, by the triangle inequality we have that\n\\begin{align*}\n& \\left|\\tilde{d}(f(x),f(y))- \\tilde{d}(f(\\pi x),f(\\pi y))\\right|\n\\\\\n& \\qquad \\leq \\frac{2\\lambda \\kappa}{m} \\sum_{i=1}^m \\left|\\mathbbm{1}_{\\{ \\operatorname{sign}((Ax)_i+\\tau_i) \\neq \\operatorname{sign} ((Ay)_i+\\tau_i)\\}} -\\mathbbm{1}_{ \\{\\operatorname{sign}((A \\pi x)_i+\\tau_i)\\neq \\operatorname{sign}((A \\pi y)_i+\\tau_i)\\}}\\right|.\n\\end{align*}\nObserve that if\n$$\n\\operatorname{sign}((Ax)_i+\\tau_i)=\\operatorname{sign}((A \\pi x)_i+\\tau_i) \\ \\ {\\rm and} \\ \\\n\\operatorname{sign}((Ay)_i+\\tau_i)=\\operatorname{sign}((A\\pi y)_i+\\tau_i),\n$$\nthen\n$$\n\\mathbbm{1}_{\\{\\operatorname{sign}( (Ax)_i+\\tau_i) \\neq \\operatorname{sign}((Ay)_i+\\tau_i)\\}} -\\mathbbm{1}_{\\{ \\operatorname{sign} ((A\\pi x)_i+\\tau_i) \\neq \\operatorname{sign}( (A \\pi y)_i+\\tau_i)\\}}=0.\n$$\nTherefore,\n\\begin{equation}\n\\label{eq:applyDM18}\n\\sup_{x,y\\in T}\\left|\\tilde{d}(f(x),f(y))- \\tilde{d}(f(\\pi x),f(\\pi y))\\right|\n\\leq \\frac{4\\lambda \\kappa}{m} \\sup_{x \\in T} \\sum_{i=1}^m \\mathbbm{1}_{\\{\\operatorname{sign}( (Ax)_i+\\tau_i)\\neq \\operatorname{sign}( (A \\pi x)_i+\\tau_i)\\}}.\n\\end{equation}\nMoreover,\n$$\n\\mathbbm{1}_{\\{\\operatorname{sign}( (Ax)_i+\\tau_i)\\neq \\operatorname{sign}( (A \\pi x)_i+\\tau_i)\\}} = \\mathbbm{1}_{\\{\\tau_i \\in [-(Ax)_i,-(A \\pi x)_i] \\cup [-(A \\pi x)_i,-(Ax)_i]\\}}\n$$\nimplying that\n\\begin{equation} \\label{eqn:indZeroSep}\n\\mathbbm{1}_{\\{\\operatorname{sign}((Ax)_i+\\tau_i)\\neq \\operatorname{sign}( (A \\pi x)_i+\\tau_i)\\}}=0\n\\end{equation}\non the set\n$$\nA_{\\delta,i} = \\left\\{x\\in \\mathbb{R}^n \\ : \\ |(A \\pi x)_i+\\tau_i|>\\delta \\geq |(A (x-\\pi x))_i|\\right\\}.\n$$\nTherefore,\n\\begin{equation} \\label{eqn:signChangeCov}\n\\mathbbm{1}_{\\{\\operatorname{sign}((Ax)_i+\\tau_i) \\neq \\operatorname{sign}( (A \\pi x)_i+\\tau_i)} \\leq \\mathbbm{1}_{A_{\\delta,i}^c} \\leq \\mathbbm{1}_{\\{|(A \\pi x)_i+\\tau_i|\\leq \\delta\\}} + \\mathbbm{1}_{\\{| (A (x-\\pi x))_i|>\\delta\\}},\n\\end{equation}\nand in particular,\n\\begin{align}\\label{eq:proof:Gaussian:twoprocesses}\n&\\sup_{x,y\\in T}\\left|\\tilde{d}(f(x),f(y))- \\tilde{d}(f(\\pi x),f(\\pi y))\\right| \\nonumber\n\\\\\n& \\qquad \\leq \\frac{4\\lambda \\kappa}{m} \\left(\\sup_{y\\in T_{\\theta}} \\sum_{i=1}^m \\mathbbm{1}_{\\{|(Ay)_i+\\tau_i|\\leq \\delta\\}}\n  + \\sup_{z\\in (T-T)\\cap \\theta B_2^n} \\sum_{i=1}^m \\mathbbm{1}_{\\{|(Az)_i|>\\delta\\}}\\right).\n\\end{align}\nClearly, for every $\\alpha \\in \\mathbb{R}$ and $1 \\leq i \\leq m$, $\\mathbb{P}_\\tau(|\\alpha+\\tau_i| \\leq \\delta) \\leq \\delta\/\\lambda$; thus,\n$$\n\\mathbb{P}_{\\tau}\\left(| (Ay)_i+\\tau_i|\\leq \\delta\\right)\\leq \\frac{\\delta}{\\lambda}.\n$$\nBy the Chernoff bound, there is an absolute constant $c_3$ such that with $\\tau$-probability at least $1-\\exp(-c_3m\\delta\/\\lambda)$,\n$$\n\\sum_{i=1}^m \\mathbbm{1}_{\\{|(Ay)_i+\\tau_i|\\leq \\delta\\}}\\leq \\frac{2\\delta m}{\\lambda}.\n$$\nHence, if\n$$\nm \\geq c_4 \\frac{\\lambda}{\\delta} \\log|T_\\theta|\n$$\nthen by the union bound, with $\\tau$-probability at least $1-\\exp(-c_3m\\delta\/(2\\lambda))$\n$$\n\\sup_{y\\in T_{\\theta}} \\frac{4\\lambda \\kappa}{m}\\sum_{i=1}^m \\mathbbm{1}_{\\{|(Ay)_i+\\tau_i|\\leq \\delta\\}} \\leq 8\\kappa \\delta.\n$$\nFinally, by \\eqref{eqn:knormoscillations}, for every $z\\in (T-T)\\cap \\theta B_2^n$,\n\\begin{equation*}\n(Az)^*_{s} \\leq \\frac{1}{\\sqrt{s}}\\|Az\\|_{[s]}\\leq \\delta,\n\\end{equation*}\nimplying that the second term on the right-hand side of \\eqref{eq:proof:Gaussian:twoprocesses} is bounded by $c\\kappa \\delta$.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\\subsection{The gaussian case} \\label{sec:gaussian}\n\nWhen $A$ is the standard gaussian matrix, the application of Theorem~\\ref{thm:mainGeneric} is straightforward. The analysis is based on two well-known facts on the way that a gaussian matrix acts on an arbitrary subset of $\\mathbb{R}^n$. The first is a consequence of the standard gaussian concentration inequality, see e.g.\\ \\cite[Lemma 2.1]{PlV14}.\n\\begin{Theorem} \\label{thm:gaussian-1}\nThere is an absolute constant $c$ such that the following holds. Let $T \\subset \\mathbb{R}^n$ and $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ be the standard gaussian matrix. Then for $u \\geq 1$,\n$$\n\\mathbb{P} \\left( \\sup_{t \\in T} \\left| \\|A t\\|_1 - \\mathbb{E} \\|At\\|_1 \\right| \\geq u\\sqrt{m} \\ell_*(T) \\right) \\leq 2\\exp\\left(-cu^2 d^*(T)\\right),\n$$\nwhere\n$$\nd^*(T) = \\left(\\frac{\\ell_*(T)}{{\\cal R}(T)}\\right)^2 \\ \\ {\\rm and} \\ \\ {\\cal R}(T)=\\sup_{t \\in T} \\|t\\|_2.\n$$\n\\end{Theorem}\nThe second fact required here is that a typical realization of $A$ maps $T$ to a set of `regular vectors'.\n\\begin{Theorem} \\label{thm:gaussian-good-position}\nThere are absolute constants $c$ and $C$ such that the following holds. Let $T \\subset \\mathbb{R}^n$ and $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ be the standard gaussian matrix. Then for $u \\geq 1$ and $1 \\leq k \\leq m$, with probability at least $1-2\\exp(-cu^2 k\\log(em\/k))$,\n$$\n\\sup_{t \\in T} \\|At\\|_{[k]} \\leq C \\left(\\ell_*(T) + u {\\cal R}(T) \\sqrt{k \\log (em\/k)} \\right).\n$$\n\\end{Theorem}\nTheorem~\\ref{thm:gaussian-good-position} follows by a standard chaining argument. We include this proof in Appendix~\\ref{sec:knormGaussian} for the sake of completeness. Using Theorem \\ref{thm:gaussian-1} and Theorem \\ref{thm:gaussian-good-position}, it is straightforward to establish Theorem \\ref{thm:main-A}.\n\n\\noindent{\\bf Proof of Theorem \\ref{thm:main-A}.} Following Theorem \\ref{thm:mainGeneric}, we apply Theorem \\ref{thm:gaussian-good-position} for the sets $T_{\\theta}$ and $(T-T) \\cap \\theta B_2^n$. Thus, for every fixed $1 \\leq k \\leq m$, with probability at least $1-2\\exp(-c_0 u^2 k \\log(em\/k))$, for every $t \\in T_{\\theta}$,\n\\begin{equation} \\label{eq:gaussian-k-norm-1}\n\\|At\\|_{[k]} \\leq c_1\\left(\\ell_*(T_{\\theta})+ u {\\cal R}(T_{\\theta}) \\sqrt{k \\log (em\/k)} \\right)\n\\end{equation}\nand for every $z \\in (T-T) \\cap \\theta B_2^n$,\n\\begin{equation} \\label{eq:gaussian-k-norm-2}\n\\|At\\|_{[k]} \\leq c_1\\left(\\ell_*((T-T)\\cap \\theta B_2^n)+ u \\theta \\sqrt{k \\log (em\/k)} \\right).\n\\end{equation}\nUsing these inequalities for $k=\\lfloor\\delta m\/\\lambda\\rfloor$, it is straightforward to verify that \\eqref{eqn:knormbias} and \\eqref{eqn:knormoscillations} hold under the stated conditions and with the stated probability. Moreover, Theorem~\\ref{thm:gaussian-1} immediately implies that \\eqref{eqn:ell1ell2assump} holds with the wanted probability for $\\kappa=\\sqrt{\\frac{\\pi}{2}}$. The result now follows from Theorem~\\ref{thm:mainGeneric}.\n\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\\begin{tcolorbox}\nInteresting as Theorem \\ref{thm:main-A} may be, it is not really a major surprise that the embedding $t \\mapsto \\operatorname{sign}(At+\\tau)$ is successful. What is far more surprising is that the seemingly unnatural estimate on the dimension happens to be sharp. The next two sections are devoted to the proof of that fact.\n\\end{tcolorbox}\n\n\\section{Proof of Theorem \\ref{thm:main-B-1}} \\label{sec:proof-B1}\n\nThe proof is based on two standard observations (see, e.g., \\cite{BLM13} for their proofs).\n\\begin{Lemma} \\label{lemma:standard}\nLet $X$ be uniformly distributed in $S^{n-1}$. Then for $0 < \\delta <1\/2$ and any $z \\in S^{n-1}$,\n$$\n\\mathbb{P}( |\\langle X,z\\rangle| \\geq \\delta  ) \\geq \\exp(-2\\delta^2 n),\n$$\nand\n$$\n\\mathbb{P}( \\|X-z\\|_2 \\leq \\alpha) \\leq \\frac{1}{\\sqrt{n}} \\alpha^{n-1}.\n$$\n\\end{Lemma}\n\nSet $T \\subset S^{n-1}$ to be a $1\/2$-separated subset of $S^{n-1}$ of cardinality  $|T| = \\exp(c_0n)$. Such a set exists thanks to a standard volumetric argument. The value of the constant $c_0$ remains unchanged for the rest of this section.\n\n\n\\vskip0.3cm\n\n\n\\begin{Lemma} \\label{lemma:first-obs}\nThere is an absolute constant $c_1$ such that the following holds. Let\n\\begin{equation} \\label{eq:cond-on-k-1}\n\\log k \\geq c_1 \\max\\{\\delta^2 n, \\log n\\}\n\\end{equation}\nand let $X_1,...,X_k$ be independent and uniformly distributed in $S^{n-1}$. Then with probability at least $0.99$, for every $t_1,t_2 \\in T$, there is a $1 \\leq j \\leq k$ such that\n$$\n|\\langle X_j,t_1-t_2\\rangle| \\geq \\delta.\n$$\n\\end{Lemma}\n\\noindent {\\bf Proof.}\\ \\  Define $U\\subset S^{n-1}$ by\n$$\nU=\\left\\{\\frac{t_i-t_j}{\\|t_i-t_j\\|_2} : t_i \\not = t_j, \\ t_i,t_j \\in T \\right\\},\n$$\nand let $\\gamma=2\\delta$. \nSince $\\|t_i-t_j\\|_2 \\geq 1\/2$ if $t_i \\not = t_j$, it suffices to show that, with the stated probability, for any $u\\in U$ there is some $1 \\leq j \\leq k$ such that\n$$\n|\\langle X_j,u\\rangle| \\geq \\gamma.\n$$\nLet $X$ be uniformly distributed in $S^{n-1}$ and set $p=\\mathbb{P}(|\\langle X,u\\rangle| \\geq \\gamma)$. By independence of the $X_j$,\n$$\n\\mathbb{P}( \\forall 1 \\leq j \\leq k, : \\ |\\langle X_j,u\\rangle| < \\gamma) = [\\mathbb{P}(|\\langle X,u\\rangle| < \\gamma)]^k = (1-p)^k\\leq e^{-kp}.\n$$\nBy the union bound, the wanted estimate follows if $|T|e^{-kp}\\leq 0.01$. Recalling that $|T| =\\exp(c_0n)$ and that, by Lemma~\\ref{lemma:standard}, $p \\geq \\exp(-c_2\\gamma^2 n)$, the condition holds when $k$ satisfies \\eqref{eq:cond-on-k-1}.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\nThe second observation we require follows immediately from the second part of Lemma~\\ref{lemma:standard} and the union bound.\n\\begin{Lemma} \\label{lemma:second-obs}\nThere are absolute constants $c_1$ and $c_2$ such that the following holds. Let $0 < \\alpha \\leq c_1$ and $\\log k \\leq c_2 n \\log(1\/\\alpha)$. Then with probability at least $0.99$, for every $1 \\leq i \\not = j \\leq k$,\n$\\|X_i \\pm X_j\\|_2 \\geq \\alpha$ and for every $1 \\leq i \\leq k$ and every $t \\in T$, $\\|X_i \\pm t\\|_2 \\geq \\alpha$.\n\\end{Lemma}\n\nCombining Lemma \\ref{lemma:first-obs} and Lemma \\ref{lemma:second-obs}, we find that there are (deterministic) sets $V,T \\subset S^{n-1}$ with the following properties:\n\\begin{description}\n\\item{$(1)$} $V\\cup(-V) \\cup T$ is an $\\alpha$-separated subset of $S^{n-1}$.\n\\item{$(2)$} For every $t_1,t_2 \\in T$ there is some $v \\in V$ such that $|\\langle v,t_1-t_2\\rangle| \\geq \\delta$.\n\\end{description}\nNext, set $s =\\frac{k}{N-1}$; split the set $V$ to $s$ disjoint sets, each  of cardinality $N-1$; and denote those sets by $V_j$, $1 \\leq j \\leq s$. Let $Y$ be uniformly distributed in $T$ and consider the random sets\n$$\nV_j \\cup \\{Y\\}, \\ \\ 1 \\leq j \\leq s,\n$$\nwhich are of cardinality $N$. Note that the only source of randomness in $V_j \\cup \\{Y\\}$ is $Y$. Let $(f,\\phi)$ be a random inner product-preserving embedding into $\\{-1,1\\}^m$ with parameters $n,\\frac{\\delta}{4},\\eta,$ and $N$ (see Definition~\\ref{def:random-emb}), and consider the event\n$$\n\\left\\{ (f,\\phi) \\ \\ \\delta\/4{\\rm -embeds \\ } V_j \\cup \\{Y\\} \\ {\\rm in \\ } \\{-1,1\\}^m \\ {\\rm for \\ every \\ } 1 \\leq j \\leq s \\right\\}.\n$$\nObserve that for every fixed realization of $Y$ and the union bound,\n$$\n\\mathbb{P}_{(f,\\phi)} \\left( \\left\\{ (f,\\phi) \\ \\ \\delta\/4{\\rm-embeds \\ } V_j \\cup \\{Y\\} \\ {\\rm in \\ } \\{-1,1\\}^m \\ {\\rm for \\ every \\ } 1 \\leq j \\leq s \\right\\} \\right) \\geq 1-s\\eta \\geq 1\/2\n$$\nprovided that\n\\begin{equation} \\label{eq:cond-on-k-11}\ns = \\frac{k}{N-1} \\leq \\frac{1}{2\\eta}.\n\\end{equation}\nIn that case,\n\\begin{align*}\n& \\mathbb{P} \\left( \\left\\{ (f,\\phi) \\ \\delta\/4{\\rm -embeds \\ } V_j \\cup \\{Y\\} \\ {\\rm in \\ } \\{-1,1\\}^m \\ {\\rm for \\ every \\ } 1 \\leq j \\leq s \\right\\} \\right)\n\\\\\n& \\qquad = \\mathbb{E}_Y \\left( \\mathbb{P}_{(f,\\phi)} \\left( \\left\\{ (f,\\phi) \\ \\delta\/4{\\rm -embeds \\ } V_j \\cup \\{Y\\} \\ {\\rm in \\ } \\{-1,1\\}^m \\ {\\rm for \\ every \\ } 1 \\leq j \\leq s \\right\\} \\right) \\big| Y \\right) \\geq \\frac{1}{2}.\n\\end{align*}\nHence, by Fubini's Theorem, there is some realization of $(f,\\phi)$ such that\n$$\n\\mathbb{P}_Y \\left( \\left\\{ (f,\\phi) \\ \\delta\/4{\\rm -embeds \\ } V_j \\cup \\{Y\\} \\ {\\rm in \\ } \\{-1,1\\}^m \\ {\\rm for \\ every \\ } 1 \\leq j \\leq s \\right\\} \\right) \\geq \\frac{1}{2}.\n$$\nThus, by the definition of $Y$, there is a set $T^\\prime \\subset T$ of cardinality at least $\\frac{1}{2}|T|=\\frac{1}{2}\\exp(c_0n)$, such that for every $t \\in T^\\prime$ and every $1 \\leq j \\leq s$, the set $V_j \\cup \\{t\\}$ is $\\delta\/4$-embedded in $\\{-1,1\\}^m$ by the fixed embedding $(f,\\phi)$.\n\n\n\n\\vskip0.3cm\nThe key observation is now as follows.\n\n\\begin{Lemma} \\label{lemma:lower-injection}\nThe embedding $f$ is injective on $T^\\prime$.\n\\end{Lemma}\n\n\\noindent {\\bf Proof.}\\ \\   Let $t_1, t_2  \\in T^\\prime$ such that $f(t_1)=f(t_2)$.\nBy Property $(2)$, there is a $v \\in V$ that `separates' $t_1$ and $t_2$; i.e.,\n$$\n\\left|\\langle t_1,v\\rangle - \\langle t_2,v\\rangle\\right| \\geq \\delta.\n$$\nLet $j$ be such that $v \\in V_j$, observe that $(f,\\phi)$ is a $\\delta\/4$-embedding of $V_j \\cup \\{t_1\\}$ and $V_j \\cup \\{t_2\\}$, and trivially we have that $\\phi(f(t_1),f(v))=\\phi(f(t_2),f(v))$. Because $(f,\\phi)$ is a $\\delta\/4$-embedding of both sets, it follows that\n\\begin{equation} \\label{eq:good-embedding}\n\\left| \\phi(f(t_1),f(v)) - \\langle t_1,v\\rangle \\right| \\leq \\frac{\\delta}{4} \\ \\ {\\rm and} \\ \\ \\left| \\phi(f(t_2),f(v)) - \\langle t_2,v\\rangle \\right| \\leq \\frac{\\delta}{4}.\n\\end{equation}\nHence,\n$$\n|\\langle t_1,v\\rangle - \\langle t_2,v\\rangle| \\leq \\left|\\langle t_1,v\\rangle-\\phi(f(t_1),f(v)) \\right| + \\left|\\langle t_2,v\\rangle-\\phi(f(t_2),f(v)) \\right| \\leq \\frac{\\delta}{2},\n$$\nwhich is a contradiction.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\nThanks to Lemma \\ref{lemma:lower-injection} we have that $2^m \\geq |T^\\prime| = \\frac{1}{2}\\exp(c_0n)$; thus, $m \\geq c_1 n$, and all that is left is to verify that it is possible to take $n$ sufficiently large. By \\eqref{eq:cond-on-k-11}, one may set $k=(N-1)\/2\\eta \\sim N\/\\eta$ and by Lemma~\\ref{lemma:first-obs} one has to ensure that\n$$\n\\log(c_1N\/\\eta) \\geq c_2 \\max\\{\\delta^2 n, \\log n\\}.\n$$\nSetting $n=c_3 \\frac{\\log(c_1N\/\\eta)}{\\delta^2}$, the wanted condition holds by the choice of $\\delta \\geq c_4(\\eta\/N)^{1\/2}$, and the condition in Lemma \\ref{lemma:second-obs} is satisfied for a well-chosen absolute constant $\\alpha$ provided that $\\delta\\leq c_5$.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\n\\section{Proof of Theorem \\ref{thm:main-B-2}} \\label{sec:proof-B2}\n\nThe proof is based on two ingredients. First, the Dvoretzky-Milman Theorem (for projections). For more information on the Dvoretzky-Milman Theorem and its central role in Asymptotic Geometry Analysis, see e.g.\\ \\cite{MR3331351}.\n\\begin{Theorem} \\label{thm:DvorMil}\nThere are absolute constants $c_1,c_2,$ and $c_3$ such that the following holds. Let $K \\subset \\mathbb{R}^n$ be a convex body, and consider $s\\leq c_1d^*(K)$. If $A:\\mathbb{R}^n \\to \\mathbb{R}^s$ is the gaussian matrix, then with probability at least $1-2\\exp(-c_2 d^*(K))$,\n$$\nc_3 \\ell_*(K) B_2^s \\subset A K.\n$$\n\\end{Theorem}\nThe proof of Theorem \\ref{thm:DvorMil} is standard. It can be found, for example, in \\cite{MR3571258}.\n\n\n\\begin{Remark}\nThe Dvoretzky-Milman Theorem actually implies a two-sided, almost isometric equivalence (i.e., $c_3$ can be made arbitrarily close to $1$, and the lower bound is complemented by an analogous upper bound). However, for our purposes, an isomorphic, one-sided bound suffices.\n\\end{Remark}\n\n\n\n\nThe second component needed here is a standard probabilistic estimate.\n\\begin{Lemma} \\label{lemma:lower-lambda}\nThere are absolute constants $c_0,c_1$ such that the following holds. Let $\\tau=(\\tau_i)_{i=1}^m$ be distributed uniformly in $[-\\lambda,\\lambda]^m$. Then for any $k \\leq m\/2$, with probability $1-c_0\\exp(-c_1k)$, there is a permutation $\\pi$ of $\\{1,...,m\\}$, such that for every $k\/2 \\leq i \\leq k$,\n$$\n\\frac{|\\tau_{\\pi(i)}|}{\\lambda} \\leq \\frac{2i}{m}.\n$$\n\\end{Lemma}\n\n\\noindent {\\bf Proof.}\\ \\  Fix $i \\leq  m\/2$. Note that $\\lambda^{-1} \\tau_j$, $j=1,\\ldots,m$, are independent and distributed uniformly in $[-1,1]$. Hence, for any $1\\leq j\\leq m$,\n$$\n\\mathbb{P}\\left( \\frac{|\\tau_j|}{\\lambda} \\leq \\frac{2i}{m}\\right) = \\frac{2i}{m}.\n$$\nBy a binomial estimate, with probability at least $1-2\\exp(-c_0 i)$,\n$$\n\\left| \\left\\{j\\in\\{1,\\ldots,m\\} : \\frac{|\\tau_j|}{\\lambda} \\leq \\frac{2i}{m} \\right\\} \\right| \\geq i.\n$$\nNow fix $k \\leq m\/2$, and the claim follows by the union bound over $k\/2 \\leq i \\leq k$.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\nLet us now complete the proof of Theorem~\\ref{thm:main-B-2}. Let $T$ be a convex body. Fix $\\delta>0$ and observe that for any $t \\in T \\cap \\delta B_2^n$ we have that $f(t)=\\operatorname{sign}(At + \\tau)$ and $f(0)=\\operatorname{sign}(\\tau)$. Therefore,\n$$\nd_H(f(t),f(0)) = \\sum_{i=1}^m \\mathbbm{1}_{\\{ \\operatorname{sign} ( (At)_i + \\tau_i) \\not = \\operatorname{sign}(\\tau_i)\\}}.\n$$\nClearly, $\\operatorname{sign} ( (A t)_i + \\tau_i) \\not = \\operatorname{sign}(\\tau_i)$ if and only if\n$$\n\\tau_i \\in [-(A t)_i,0) \\cup [0,-(A t)_i).\n$$\n\nLet $k \\leq m\/2$ to be specified in what follows. By Lemma~\\ref{lemma:lower-lambda}, with probability at least $1-2\\exp(-c_1 k)$ there exists a permutation $\\pi$ of $\\{1,...,m\\}$ (depending on the realization of $(\\tau_i)_{i=1}^m$) for which\n$$\n\\frac{|\\tau_{\\pi(i)}|}{\\lambda} \\leq \\frac{2i}{m} \\ \\ \\ {\\rm for \\ every } \\ \\ \\frac{k}{2} \\leq i \\leq k.\n$$\nConditioned on that event (which depends only on $\\tau$), there is an absolute constant $c_2$ such that\n$$\n\\left(\\sum_{i=k\/2}^{k} \\tau_{\\pi(i)}^2 \\right)^{1\/2} \\leq c_2 \\lambda \\frac{k^{3\/2}}{m}.\n$$\nLet $I=\\{ \\pi(i) : k\/2 \\leq i \\leq k\\}$ and let $P_I$ be the projection operator onto ${\\rm span}(e_i)_{i \\in I}$. Then $P_I A$ is a $\\frac{k}{2} \\times n$ standard gaussian matrix, and by Theorem \\ref{thm:DvorMil}, with probability at least\n$$\n1-2\\exp(-c_3d^*(T \\cap \\delta B_2^n)),\n$$\nwe have that $P_I A (T \\cap \\delta B_2^n)$ contains the Euclidean ball\n$$\nc_4 \\ell_*(T \\cap \\delta B_2^n) B_2^{|I|}\n$$\nas long as\n\\begin{equation}\n\\label{eqn:condonk1}\nk \\leq c_5 d^*(T \\cap \\delta B_2^n) \\sim \\frac{\\ell_*^2(T \\cap \\delta B_2^n)}{\\delta^2}.\n\\end{equation}\nHence, if in addition\n\\begin{equation}\n\\label{eqn:condonk2}\nc_2 \\lambda \\frac{k^{3\/2}}{m} \\leq c_4 \\ell_*(T \\cap \\delta B_2^n),\n\\end{equation}\nthen there is some $t^* \\in T \\cap \\delta B_2^n$ such that for every $i \\in I$\n$$\n\\tau_i \\in [-(A t^*)_i,0) \\cup [0,-(A t^*)_i).\n$$\nIn particular,\n$$\nd_H(f(t^*),f(0)) \\geq \\frac{k}{2} \\ \\ \\  \\text{and} \\ \\ \\  \\|t^*-0\\|_2 \\leq \\delta.\n$$\nThe required conditions \\eqref{eqn:condonk1}, \\eqref{eqn:condonk2}, and $k\\leq m\/2$ are all satisfied for\n$$\nk \\sim \\min\\left\\{ \\left(\\frac{m}{\\lambda} \\ell_*(T \\cap \\delta B_2^n)\\right)^{2\/3}, \\frac{\\ell_*^2(T \\cap \\delta B_2^n)}{\\delta^2}\\right\\},\n$$\nthanks to the assumed lower bound on $\\lambda$. \n\nTherefore,\n$$\n\\frac{\\lambda}{m} d_H(f(t^*),f(0)) \\geq \\frac{\\lambda k}{2m} \\geq c_6 \\lambda \\min\\left\\{ \\frac{\\ell_*^{2\/3}(T \\cap \\delta B_2^n)}{\\lambda^{2\/3} m^{1\/3}},\\frac{\\ell_*^2(T \\cap \\delta B_2^n)}{m \\delta^2}\\right\\}.\n$$\nand if\n$$\nm \\leq c_7\\lambda \\frac{\\ell_*^2(T \\cap \\delta B_2^n)}{\\delta^3}\n$$\nit follows that\n$$\n\\left| \\sqrt{2\\pi} \\frac{\\lambda}{m} d_{H}(f(t^*),f(0)) - \\|t^*-0\\|_2 \\right| \\geq 2\\delta.\n$$\nThe claim is now evident by noticing that if $T$ is a convex body, then $T-T = 2T$.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\\subsection{The lower bound on $\\lambda$}\nThe lower bound in Theorem \\ref{thm:main-B-2} comes with a caveat: that\n$$\n\\lambda \\sqrt{m} \\gtrsim \\ell_*((T-T) \\cap \\delta B_2^n).\n$$\nLet us show that this caveat is not very restrictive.\n\\begin{Lemma} \\label{lemma:lower-on-lambda}\nThere is an absolute constant $c$ and for every $\\beta>0$ there is a constant  $\\gamma=\\gamma(\\beta) \\geq 1$ such that the following holds.\nAssume that\n\\begin{equation} \\label{eq:in-lemma-lower-on-lambda}\n\\ell_*( (T-T) \\cap \\lambda B_2^n) > \\gamma \\ell_*( (T-T) \\cap \\delta B_2^n).\n\\end{equation}\nIf, with probability at least $0.9$, the mapping $t \\mapsto \\operatorname{sign}(At+\\tau)$ is a $\\delta$-embedding of a convex body $T$, then\n\\begin{equation}\n\\label{eq:in-lemma-lower-on-lambda-conclusion}\n\\beta \\lambda \\sqrt{m} \\geq \\ell_*((T-T)\\cap \\delta B^n_2).\n\\end{equation}\n\\end{Lemma}\nIt follows that if the function $r \\mapsto \\ell_*( (T-T) \\cap r B_2^n)$ exhibits a regular decay around $r=\\delta$, then the condition in Lemma \\ref{lemma:lower-on-lambda} is satisfied once $\\lambda \\gtrsim \\delta$. As is explained in Appendix~\\ref{sec:minShift}, the stronger condition $\\lambda\\gtrsim {\\cal R}(T)$ is needed for the mapping $t \\mapsto \\operatorname{sign}(At+\\tau)$ to have any chance of separating close points on rays $\\{\\alpha x : \\alpha \\geq 0\\}$.\n\n\\begin{Remark}\nThe proof of Theorem \\ref{thm:main-B-2} shows that the value of $\\beta$ that is needed is just an absolute constant. Therefore, $\\gamma$ in Lemma \\ref{lemma:lower-on-lambda} may be taken to be an absolute constant as well.\n\\end{Remark}\n\n\\vskip0.3cm\nThe proof of Lemma \\ref{lemma:lower-on-lambda} is based on the behaviour of the function\n$$\n\\psi(r)=\\frac{1}{\\sqrt{m}r}\\ell_*( (T-T) \\cap r B_2^n).\n$$\nOn the one hand, the map $r \\mapsto \\ell_*( (T-T) \\cap r B_2^n)$ is increasing. On the other hand, it is standard to verify that $\\psi$ is continuous and decreasing in $(0,\\infty)$. Also, it tends to $0$ at infinity and to $\\sqrt{n\/m}$ at $0$ (because $T-T=2T$ is a convex body and in particular has a nonempty interior). Also note that by \\eqref{eq:in-lemma-lower-on-lambda}, $\\delta < \\lambda\/\\gamma$. Indeed, \\eqref{eq:in-lemma-lower-on-lambda} and the monotonicity of $r \\mapsto \\ell_*( (T-T) \\cap r B_2^n)$ imply that $\\delta \\leq \\lambda$ (as $\\gamma\\geq 1$). The monotonicity of $\\psi$ therefore implies that $\\psi(\\lambda)\\leq \\psi(\\delta)$ and hence\n$$\n\\ell_*((T-T) \\cap \\lambda B_2^n) \\leq \\frac{\\lambda}{\\delta} \\ell_*((T-T) \\cap \\delta B_2^n) < \\frac{\\lambda}{\\delta} \\cdot \\frac{1}{\\gamma} \\ell_*((T-T) \\cap \\lambda B_2^n).\n$$\nThe monotonicity of $\\psi$ and the fact that $\\delta<\\lambda\/\\gamma$ are used frequently in the proof of Lemma~\\ref{lemma:lower-on-lambda}.\n\n\n\n\n\\noindent {\\bf Proof.}\\ \\   Let $c_0$ be an absolute constant whose value is specified in what follows.  The key to the proof is to show that if $\\rho$ satisfies\n\\begin{equation} \\label{eq:lower-fixed-1}\n\\ell_*((T-T)\\cap \\delta B_2^n) \\leq c_0 \\frac{\\rho}{\\gamma} \\sqrt{m},\n\\end{equation}\nand\n\\begin{equation} \\label{eq:lower-fixed-2}\n\\ell_*((T-T)\\cap \\rho B_2^n) \\geq  c_0 \\rho \\sqrt{m}\n\\end{equation}\nthen $\\lambda \\geq c^\\prime \\rho$ for an absolute constant $c^\\prime$. Once we establish that, the claim follows immediately: by \\eqref{eq:lower-fixed-1}, we have that\n$$\n\\ell_*((T-T)\\cap \\delta B^n_2) \\leq c_0\\frac{\\rho}{\\gamma} \\sqrt{m} \\leq  \\frac{c_0}{c^\\prime \\gamma} \\lambda\\sqrt{m} \\leq \\beta \\lambda \\sqrt{m},\n$$\nprovided that $\\gamma \\geq c_0\/(c^\\prime \\beta)$.\n\n\nThe starting point of the proof is, once again, the Dvoretzky-Milman Theorem. By \\eqref{eq:lower-fixed-2}, the Dvoretzky-Milman dimension of $(T-T)\\cap \\rho B_2^n$ satisfies\n$$\nd^*((T-T)\\cap \\rho B_2^n) \\geq c_0^2 m.\n$$\nIn particular, if $c_0$ is a sufficiently large absolute constant and $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ is the standard gaussian matrix, then with probability at least $0.9$,\n$$\nc_1 \\ell_*((T-T)\\cap \\rho B_2^n) B_2^m \\subset A((T-T)\\cap \\rho B_2^n)\n$$\nfor an absolute constant $c_1$. For the rest of the proof, the values of $c_0$ and $c_1$ remain unchanged.\n\nBefore proceeding to the heart of the proof, let us show that we may assume without loss of generality that a suitable choice of $\\rho$, satisfying \\eqref{eq:lower-fixed-1} and \\eqref{eq:lower-fixed-2}, exists.\n\nConsider two cases: if $c_0 \\geq \\sqrt{n\/m}$ then trivially, for any $r>0$, $\\psi(r) \\leq c_0$. In other words, for every $r>0$,\n$$\n\\ell_*( (T-T) \\cap r B_2^n) \\leq c_0 r \\sqrt{m}.\n$$\nIn particular, for $r=\\delta$,\n$$\n\\ell_*( (T-T) \\cap \\delta B_2^n) \\leq c_0 \\delta \\sqrt{m} \\leq \\beta \\lambda \\sqrt{m},\n$$\nwhen $\\lambda \\geq c_0\\delta\/\\beta$. Recall that $\\lambda > \\gamma \\delta$, and thus it suffices that $\\gamma \\geq c_0\/\\beta$ for the wanted estimate \\eqref{eq:in-lemma-lower-on-lambda-conclusion} to hold.\n\nTurning to the second case, when $c_0 \\leq \\sqrt{n\/m}$, then since $\\psi$ is continuous and decreasing from $\\sqrt{n\/m}$ to $0$, there is some $\\rho^*$ for which\n\\begin{equation}\n\\label{eqn:rhoStarDef}\n\\ell_*((T-T)\\cap \\rho^* B_2^n) = c_0\\rho^* \\sqrt{m}.\n\\end{equation}\nIn particular, $\\rho^*$ satisfies \\eqref{eq:lower-fixed-2}.\n\nNext, observe that either $\\rho^*$ satisfies \\eqref{eq:lower-fixed-1} and also $\\lambda \\leq \\rho^*$, or, alternatively, the wanted estimate \\eqref{eq:in-lemma-lower-on-lambda-conclusion} holds. Indeed, if $\\lambda \\geq \\rho^*$ there is nothing to prove: in that case, using our assumption \\eqref{eq:in-lemma-lower-on-lambda}, $\\psi(\\lambda)\\leq \\psi(\\rho^*)$, and \\eqref{eqn:rhoStarDef} we find\n$$\n\\ell_*( (T-T) \\cap \\delta B_2^n) < \\frac{1}{\\gamma} \\ell_*( (T-T) \\cap \\lambda B_2^n) \\leq \\frac{c_0}{\\gamma} \\lambda \\sqrt{m} \\leq \\beta \\lambda \\sqrt{m}\n$$\nprovided that $\\gamma \\geq c_0\/\\beta$. On the other hand, consider the case $\\lambda \\leq \\rho^*$. If $\\rho^*$ does not satisfy \\eqref{eq:lower-fixed-1} then by \\eqref{eqn:rhoStarDef} and \\eqref{eq:in-lemma-lower-on-lambda},\n$$\n\\ell_*((T-T)\\cap \\delta B_2^n) > c_0\\frac{\\rho^*}{\\gamma} \\sqrt{m} = \\frac{1}{\\gamma} \\ell_*((T-T)\\cap \\rho^* B_2^n) \\geq \\frac{1}{\\gamma} \\ell_*((T-T)\\cap \\lambda B_2^n) > \\ell_*((T-T)\\cap \\delta B_2^n),\n$$\nwhich is impossible.\n\nTherefore, we may assume for the remainder of the proof that $c_0 \\leq \\sqrt{n\/m}$, that $\\rho^*$ satisfies both \\eqref{eq:lower-fixed-1} and \\eqref{eq:lower-fixed-2}, and that $\\lambda \\leq \\rho^*$. To complete the proof we require three observations.\n\nFirst, $\\delta \\leq \\rho^*\/\\gamma$ since $\\delta \\leq \\lambda \\leq \\rho^*$ and $\\delta \\leq \\lambda\/\\gamma$.\n\nSecond, by applying Theorem~\\ref{thm:gaussian-good-position} (with $k=m$) to the set\n$$\n(T-T) \\cap \\delta B_2^n=2T \\cap \\delta B_2^n,\n$$\nand using \\eqref{eq:lower-fixed-1}, there is an absolute constant $c_2$ such that with probability at least $0.99$, for any $t \\in T$,\n$$\nA\\left( \\left(t + \\{z \\in T : \\|t-z\\|_2 \\leq \\delta\\} \\right) \\right) \\subset At + c_2 \\left(\\delta+\\frac{\\rho^*}{\\gamma}\\right) \\sqrt{m} B_2^m \\subset At + c_3 \\frac{\\rho^*}{\\gamma} \\sqrt{m} B_2^m,\n$$\nwhere we have used that $\\delta \\leq \\rho^*\/\\gamma$.\n\nThird, as noted previously, by the choice of $c_0$ and the Dvoretzky-Milman Theorem, with probability at least $0.99$,\n$c_1 \\ell_*((T-T)\\cap \\rho^* B_2^n) B_2^m \\subset A((T-T)\\cap \\rho^* B_2^n)$. Moreover, $T-T=2T$ and $2T \\cap \\rho^* B_2^n = 2(T \\cap (\\rho^*\/2)B_2^n)$. Therefore,\n$$\nc_1\\cdot c_0 \\rho^* \\sqrt{m} B_2^m = c_1 \\ell_*((T-T)\\cap \\rho^* B_2^n) B_2^m \\subset 2 A(T \\cap (\\rho^*\/2) B_2^n).\n$$\nNow set $c_4=\\frac{c_1}{2}c_0$ and note that $c_4$ is an absolute constant. If $\\tau$ is supported in the interval $[-c_4\\rho^*\/10,c_4 \\rho^*\/10]$, then any two points in the same ``quadrant\" of the set\n$$\nV=c_4 \\rho^* \\sqrt{m} B_2^{m} \\cap \\left\\{ |x_i| \\geq \\frac{c_4 \\rho^*}{10}, {\\rm for \\ every \\ } 1 \\leq i \\leq m \\right\\},\n$$\nare indistinguishable for any realization of the function $y \\mapsto \\operatorname{sign}(y+\\tau)$. Clearly, $V \\subset A(T \\cap (\\rho^*\/2) B_2^n)$ and the diameter of each of its quadrants is at least $(c_4\/10) \\rho^* \\sqrt{m}$.\n\nFix a realization of $A$ in the intersection of the two events stated above---which has probability at least $0.98$. On that event, for every $t \\in T$,\n$$A\\left( \\left(t + \\{z \\in T : \\|t-z\\|_2 \\leq \\delta\\} \\right) \\right)$$\nhas diameter at most $2c_3 \\frac{\\rho^*}{\\gamma} \\sqrt{m}$. On the other hand, any quadrant ${\\cal Q}$ of $V$ has diameter at least $(c_4\/10) \\rho^* \\sqrt{m}$. Hence, we can pick $y_1,y_2$ in ${\\cal Q}$ with $\\|y_1-y_2\\|_2\\geq (c_4\/10) \\rho^* \\sqrt{m}$. Since $V \\subset A(T \\cap (\\rho^*\/2) B_2^n)$, $y_1=A t_1$ and $y_2=A t_2$ for some $t_1,t_2\\in T$. If $\\gamma$ is large enough so that \n$$2c_3 \\frac{\\rho^*}{\\gamma}<(c_4\/10) \\rho^*$$  \nthen $y_1,y_2$ cannot be contained in $A\\left(t + \\{z \\in T: \\|t-z\\|_2 \\leq \\delta\\} \\right)$ for a single $t \\in T$; in particular, $\\|t_1-t_2\\|_2>\\delta$. Finally, if $\\lambda\\leq c_4 \\rho^*\/10$, then since $y_1,y_2$ belong to the same quadrant of $V$ it would imply that $\\operatorname{sign}(At_1+\\tau)=\\operatorname{sign}(At_2+\\tau)$, violating the assumption that $t \\mapsto \\operatorname{sign}(At+\\tau)$ is a $\\delta$-embedding of $T$ with probability at least $0.9$. Thus, we must have $\\lambda \\geq \\frac{c_4}{10}\\rho^*$, and $c_4\/10$ is an absolute constant---as required.    \n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\n\\subsection{A counterexample}\nNext, let us explore the conjecture that $m \\sim \\ell_*^2(T)\/\\delta^2$ suffices to ensure that $f(t)=\\operatorname{sign}(At+\\tau)$ is, with high probability, a $\\delta$-embedding of an arbitrary $T \\subset \\mathbb{R}^n$. We show that this conjecture is false, and that sometimes one in fact needs $m \\geq c\\ell_*^2(T)\/\\delta^3$.\n\nFix $0<\\varepsilon<\\delta\/2$, let $r \\in \\mathbb{N}$, set $I=\\{1,...,r\\}$ and put $J=\\{r+1,...,\\eta r\/\\varepsilon^2\\}$ for an absolute constant $\\eta$ to be specified in what follows. Let $n=|I|+|J|$ and denote by $B_2^I$ and $B_2^J$ the Euclidean balls supported on ${\\rm span}(e_i, i \\in I)$ and ${\\rm span}(e_j : j \\in J)$ respectively.\nSet\n$$\n{\\cal E}=B_2^I \\otimes \\varepsilon B_2^J = \\left\\{ (x,y): x \\in B_2^I, \\ y \\in \\varepsilon B_2^J \\right\\} \\subset \\mathbb{R}^{n},\n$$\nand note that ${\\cal E}$ is a convex body. \n\nConsider our random embedding $f:\\mathbb{R}^n \\to \\{-1,1\\}^m$, $f(t)=\\operatorname{sign}(At+\\tau)$. Observe that for any $0<\\delta<\\frac{1}{4}$, ${\\cal E}$ contains two points $x,y$ located on a ray emanating from the origin such that $\\|x\\|_2=1$, $\\|y\\|_2<\\|x\\|_2$, and $\\|x-y\\|_2=2\\delta$. As we show in detail in Appendix~\\ref{sec:minShift}, this causes $f$ to fail to be a $\\delta$-embedding of ${\\cal E}$ with probability at least $0.9$ if $\\lambda\\leq c$, where $c$ is an absolute constant. Thus, to have any hope of obtaining a $\\delta$-embedding, one needs to assume that $\\lambda>c$, and $c$ is of the order of the diameter of ${\\cal E}$. We will make a slightly stronger assumption: that $\\lambda \\geq C \\sqrt{\\log(e\/\\delta)}$; it corresponds to the context of Theorem~\\ref{thm:main-A}.\n\nLet us now assume that $m$ is such that, with probability at least $0.9$, $f$ is a $\\delta$-embedding of ${\\cal E}$. Using the notation of the previous section, the proof of Theorem \\ref{thm:main-B-2} shows that for an absolute constant $\\beta$, $\\lambda$ must satisfy that\n\\begin{equation} \\label{eq:in-counter-1}\n\\beta \\lambda \\sqrt{m} \\geq \\ell_*( ({\\cal E}-{\\cal E}) \\cap \\delta B_2^n).\n\\end{equation}\nBy Lemma \\ref{lemma:lower-on-lambda}, it suffices to verify that\n\\begin{equation} \\label{eq:in-counter-2}\n\\ell_*(({\\cal E}-{\\cal E}) \\cap \\lambda B_2^n) > \\gamma(\\beta) \\ell_*(({\\cal E}-{\\cal E}) \\cap \\delta B_2^n)\n\\end{equation}\nfor a constant $\\gamma=\\gamma(\\beta)$ (and thus, an absolute constant as well) to ensure that \\eqref{eq:in-counter-1} holds.\n\n\nWithout loss of generality, we have that $\\lambda \\geq {\\cal R}({\\cal E})=\\sup_{t \\in {\\cal E}} \\|t\\|_2=1$, and to verify \\eqref{eq:in-counter-2} it is enough to ensure that\n\\begin{equation} \\label{eq:in-counter-3}\n\\ell_*(2{\\cal E}) \\geq \\gamma(\\beta) \\ell_*( 2{\\cal E} \\cap \\delta B_2^n).\n\\end{equation}\nClearly, \\eqref{eq:in-counter-3} holds if $\\ell_*(B_2^I) \\geq \\gamma(\\beta) (\\ell_*(\\delta B_2^I)+\\ell_*(2\\varepsilon B_2^J))$, i.e., that \n$$\n\\sqrt{r} \\geq c \\gamma(\\beta) (\\delta\\sqrt{ r}+\\varepsilon \\sqrt{r}\\sqrt{1+\\eta\/\\varepsilon^2}).\n$$ \nIt follows that if $\\delta$ and $\\eta$ are sufficiently small (depending on the absolute constant $\\gamma(\\beta)$), the wanted estimate holds.\n\nNow we may assume that $\\eta$ is a suitable, sufficiently small absolute constant and that $\\delta$ is also small enough---ensuring that \\eqref{eq:in-counter-3} holds. Set $\\theta \\sim \\delta\/\\log(e\\lambda\/\\delta)$, and by the choice of $\\lambda$ we have that $\\theta \\sim \\delta\/\\log(e\/\\delta)$.\n\n\n\\begin{Theorem} \\label{thm:counter-example}\nThere are absolute constants $c_0$,$c_1$, and $c_2$ such that the following holds.\nLet $\\delta\\leq c_0$ and let $A:\\mathbb{R}^n \\to \\mathbb{R}^m$ be the standard gaussian matrix. If $m \\leq c_1 \\sqrt{\\log(e\/\\delta)}\\frac{\\ell_*^2({\\cal E})}{\\delta^3}$,\nthen with probability at least $0.9$ there are $x,y \\in {\\cal E}$ such that\n$$\n\\left| \\sqrt{2\\pi} \\frac{\\lambda}{m} d_{H}(f(x),f(y)) - \\|x-y\\|_2 \\right| \\geq 2\\delta.\n$$\nMoreover, if $m \\geq c_2 \\sqrt{\\log(e\/\\delta)}\\frac{\\ell_*^2({\\cal E})}{\\delta^3}$ then with probability at least 0.9\n$$\n\\sup_{x,y \\in {\\cal E}} \\left| \\sqrt{2\\pi} \\frac{\\lambda}{m} d_{H}(f(x),f(y)) - \\|x-y\\|_2 \\right| \\leq \\delta.\n$$\n\\end{Theorem}\n\n\\noindent {\\bf Proof.}\\ \\  Set $\\varepsilon=\\theta\/2$ and let $U \\subset B_2^I\\otimes 0$ be a minimal $\\theta\/2$-cover of $B_2^I$ with respect to the Euclidean norm. Thus, $U$ is also a $\\theta$-cover of ${\\cal E}$: every $x \\in  {\\cal E}$ is supported on $I \\cup J$, and there is some $u \\in U$ such that\n$$\n\\|x-u\\|_2 = \\|P_I x -u\\|_2 + \\|P_Jx\\|_2 \\leq \\theta\/2 + \\varepsilon \\leq \\theta.\n$$\nMoreover, by a standard volumetric estimate,\n$$\n\\log {\\cal N}({\\cal E}, \\theta) \\leq \\log {\\cal N}(B_2^I, \\theta\/2) \\lesssim r \\log(5\/\\theta).\n$$\nNext, note that $\\ell_*({\\cal E})$, $\\ell_*(B_2^I)$ and $\\ell_*(\\varepsilon B_2^J)$ are all equivalent to $\\sqrt{r}$. Moreover, using the choices of $\\varepsilon$ and $\\theta$, it is straightforward to verify that there is an absolute constant $c_1$ such that \n$$\nc_1\\ell_*({\\cal E}) \\leq \\ell_*(\\varepsilon B_2^J) \\leq \\ell_*({\\cal E} \\cap \\theta B_2^n) \\leq \\ell_*({\\cal E} \\cap \\delta B_2^n) \\leq \\ell_*({\\cal E}).\n$$\nHence, by Theorem \\ref{thm:main-B-2}\n$$\nm \\geq c_2 \\sqrt{\\log(e\/\\delta)} \\frac{\\ell_*^2({\\cal E})}{\\delta^3}\n$$\nis a necessary condition for ensuring that with high probability, $t \\mapsto \\operatorname{sign}(At+\\tau)$ is a $\\delta$-embedding of ${\\cal E}$ in $\\{-1,1\\}^m$. \n\n\nAt the same time, recalling that $\\lambda\\sim \\sqrt{\\log(e\/\\delta)}$, we see that the second term in the upper bound \\eqref{eqn:main-ABdm} from Theorem \\ref{thm:main-A} (featuring the gaussian mean-width) is dominant for $\\delta$ that is sufficiently small.  Thus, the lower bound is matched by the upper one from Theorem \\ref{thm:main-A}.\n\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak}\n\n\\begin{Remark}\n(Possible logarithmic improvement) If $c<\\lambda\\leq C \\sqrt{\\log(e\/\\delta)}$, then Theorem~\\ref{thm:main-A} is not applicable for $T={\\cal E}$. However, Theorem \\ref{thm:main-B-2} shows in this context that $f$ fails to be a $\\delta$-embedding with probability at least $0.9$ if $m \\leq c\\frac{\\ell_*^2({\\cal E})}{\\delta^3}$. We leave the question whether Theorem \\ref{thm:main-A} continues to hold if $\\lambda\\geq C' {\\cal R}(T)$ unanswered. If the answer is positive, then that would yield matching bounds for $T={\\cal E}$ in the slightly improved setting $\\lambda \\sim {\\cal R}({\\cal E})$.\n\\end{Remark}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}