bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

analytic torsion of a bounded generalized cone we compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of m spreafico and using the symmetry in the de rham complex as established by m lesch we evaluate our result in lower dimensions and further provide a separate computation of analytic torsion of a bounded generalized cone over s 1 since the standard cone over the sphere is simply a flat disc

bulk universality of general β ensembles with non convex potential we prove the bulk universality of the ensembles with non convex regular analytic potentials for any 0 this removes the convexity assumption appeared in the earlier work p bourgade l erdos and h t yau universality of general ensembles preprint arxiv 0907 5605 2011 the convexity condition enabled us to use the logarithmic sobolev inequality to estimate events with small probability the new idea is to introduce a convexified measure so that the local statistics are preserved under this convexification

zeta regularized determinants for conic manifolds we study relative zeta regularized determinants of laplace type operators on compact conic manifolds we establish gluing formulae for relative zeta regularized determinants for arbitrary self adjoint extensions of the laplace beltrami operator we express the relative determinants for these as a ratio of the determinants of certain finite matrices for the self adjoint extensions corresponding to dirichlet and neumann conditions the formula is particularly simple and elegant

regularity conditions in the clt for linear eigenvalue statistics of wigner matrices abstract we show that the variance of centred linear statistics of eigenvalues of gue matrices remains bounded for large n for some classes of test functions less regular than lipschitz functions this observation is suggested by the limiting form of the variance which has previously been computed explicitly but it does not seem to appear in the literature we combine this fact with comparison techniques following tao vu and erdos yau et al and a littlewood paley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the holder class c 1 2 in the case of matrices of gaussian convolution type we also give a variance bound which implies the clt for test functions in the sobolev space h 1 and c 1 for general wigner matrices satisfying moment conditions if the additional assumption of the test function being supported away from the edge of the spectrum is made we prove the clt for test functions of regularity h 1 2 l and h 1 2 for gue and johansson matrices respectively previous results on the clt impose the existence and continuity of at least one classical derivative

witt spaces a geometric cycle theory for ko homology at odd primes introduction this paper studies a class of stratified piecewiselinear pseudomanifolds which we call witt spaces characterized by natural local intersection homology conditions 11 we compute the cobordism groups by introducing an invariant taking values in the witt group of symmetric bilinear forms over the rationals w q these pseudomanifolds solve a problem posed by d sullivan 25 to construct a class of p l cycles with signature which represent the connected version of ko homology at odd primes ko 0d z 1 2 the result was one of the first applications of the intersection homology theory of goresky and macpherson 11 specifically the rational intersection homology groups of witt spaces satisfy a poincare duality theorem to each witt space x we associate a p l invariant w x with values in w q this invariant generalizes the signature of manifolds and satisfies cobordism invariance additivity and a product formula adapting classical surgery spherical modification to this setting we prove that w x determines the cobordism class of x and obtain an explicit description of the witt cobordism groups the only nontrivial groups occur in dimensions 4k and for k 0 they are w q sullivan 25 27 has shown that a class of p l pseudomanifolds equipped with such an invariant possesses canonical orientations in ko 0 z 1 2 the orientations induce a natural transformation it of homology theories from witt space bordism to ko 0 z 1 2 the structure of w q is known 14 and we conclude that it is an equivalence at odd primes in this paper we study the cobordism theory of a class of stratified p l pseudomanifolds by means of the recent intersection homology theory of goresky and macpherson 11 goresky and macpherson were interested in finding a class of spaces with cobordism invariant signature for the purpose of extending the hirzebruch l class to the setting of manifolds with

convergence of a semi lagrangian scheme for the one dimensional vlasov poisson system a semi lagrangian scheme is proposed for solving the periodic one dimensional vlasov poisson system in phase space on unstructured meshes the distribution function f t x v and the electric field e t x are shown to converge to the exact solution values in the l infty norm the rate of convergence is in o h4 3

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

the convergence theory of particle in cell methods for multidimensional vlasov poisson systems in this work a convergence theory of particle in cell methods for multidimensional vlasov poisson systems is provided such methods replace the continuous distribution of masses or charges by disc

the resolvent expansion for second order regular singular operators abstract the paper gives a scheme for computing the asymptotics of tr e tl as t 0 where l is an elliptic operator of the form l d 2 x 2 a x and a x is a family of operators satisfying appropriate ellipticity and smoothness conditions a principle example is the laplace operator for a manifold with an asymptotically conic singularity the expansion has the usual terms away from the singularity appropriately regularized at x 0 plus singular contributions determined by the function of a 0 1 4 1 2 applications to index theorems are given in a subsequent paper

regular singular sturm liouville operators and their zeta determinants abstract we consider sturm liouville operators on the line segment 0 1 with general regular singular potentials and separated boundary conditions we establish existence and a formula for the associated zeta determinant in terms of the wronski determinant of a fundamental system of solutions adapted to the boundary conditions this generalizes the earlier work of the first author treating general regular singular potentials but only the dirichlet boundary conditions at the singular end and the recent results by kirsten loya park for general separated boundary conditions but only special regular singular potentials

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

fixed energy universality for generalized wigner matrices we prove the wigner dyson mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and hermitian wigner matrices previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for hermitian matrices if the energy parameter is fixed we develop a homogenization theory of the dyson brownian motion and show that microscopic universality follows from mesoscopic statistics 2015 wiley periodicals inc

dyson brownian motion for general beta and potential at the edge in this paper we compare the solutions of the dyson brownian motion for general beta and potential v and the associated mckean vlasov equation near the edge under suitable conditions on the initial data and the potential v we obtain optimal rigidity estimates of particle locations near the edge after a short time t mathrm o 1 our argument uses the method of characteristics along with a careful estimate involving an equation of the edge with the rigidity estimates as an input we prove a central limit theorem for mesoscopic statistics near the edge which as far as we know has been done for the first time in this paper additionally combining our results with landon and yau edge statistics of dyson brownian motion arxiv 1712 03881 2017 we give a proof of the local ergodicity of the dyson brownian motion for general beta and potential at the edge i e we show the distribution of extreme particles converges to the tracy widom beta distribution in a short time

the exotic heat trace asymptotics of a regular singular operator revisited we discuss the exotic properties of the heat trace asymptotics for a regular singular operator with general boundary conditions at the singular end as observed by falomir muschietti pisani and seeley unusual poles of the functions for some regular singular differential operators j phys a 36 39 9991 10010 2003 10 1088 0305 4470 36 39 302 as well as by kirsten loya and park the very unusual properties of the resolvent heat kernel and zeta function for the operator d2 dr2 1 4r2 j math phys 47 4 043506 2006 10 1063 1 2189194 we explain how their results alternatively follow from the general heat kernel construction by mooers heat kernel asymptotics on manifolds with conic singularities j anal math 78 1 36 1999 10 1007 bf02791127 a natural question that has not been addressed yet as the latter work did not elaborate explicitly on the singular structure of the heat trace expansion beyond the statement of non polyhomogeneity of the heat kernel

fixed energy universality of dyson brownian motion abstract we consider dyson brownian motion for classical values of with deterministic initial data v we prove that the local eigenvalue statistics coincide with the goe gue in the fixed energy sense after time t 1 n if the density of states of v is bounded above and below down to scales t in a window of size l t our results imply that fixed energy universality holds for essentially any random matrix ensemble for which averaged energy universality was previously known our methodology builds on the homogenization theory developed in 16 which reduces the microscopic problem to a mesoscopic problem as an auxiliary result we prove a mesoscopic central limit theorem for linear statistics of various classes of test functions for classical dyson brownian motion

convergence of a vortex in cell method for the two dimensional euler equations on decrit une methode des tourbillons dans les cellules pour les equations d euler bidimensionnelles on etablit une estimation d erreur

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

convergence of a high order semi lagrangian scheme with propagation of gradients for the one dimensional vlasov poisson system in this paper we give a proof of convergence of a new numerical method introduced in n besse and e sonnendrucker j comput phys 191 2003 pp 341 376 for the vlasov equation the numerical method is based on the semi lagrangian principle and the transport of the gradient of the statistical distribution function in order to get a high order and stable reconstruction these kinds of new schemes have been successfully implemented on unstructured meshes of four dimensional phase space cf n besse etude mathematique et numerique de l equation de vlasov sur des maillages non structures de l espace des phases these de l universite louis pasteur strasbourg france 2003 n besse j segre and e sonnendrucker transport theory statist phys 34 2005 pp 311 332 in order to make a rigorous proof of convergence of this method and simplify the convergence analysis we have considered the periodic one dimensional vlasov poisson system in phase space on a grid the distribution f t x v and the electric field are shown to converge to the exact solution values in h 1 norm the rate of convergence is of mathcal o delta t 2 frac delta x 4 alpha delta t frac delta v 4 alpha delta t alpha in n 2 alpha leq 1

beta ensembles stochastic airy spectrum and a diffusion we prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low lying eigenvalues of the random schrodinger operator d2 dx2 x 2 bx restricted to the positive half line where b x is white noise in doing so we extend the definition of the tracy widom distributions to all 0 and also analyze their tails last in a parallel development we provide a second characterization of these laws in terms of a one dimensional diffusion the proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit in particular we show how tracy widom laws arise from a functional central limit theorem

the wigner semi circle law and eigenvalues of matrix valued diffusions a system of stochastic differential equations for the eigenvalues of a symmetric matrix whose components are independent ornstein uhlenbeck processes is derived this corresponds to a diffusion model of an interacting particles system with linear drift towards the origin and electrostatic inter particle repulsion the associated empirical distribution of particles is shown to converge weakly as the number of particles tends to infinity to a limiting measure valued process which may be characterized as the weak solution of a deterministic ode the wigner semi circle density is found to be one of the equilibrium points of this limiting equation

zeta determinants for regular singular laplace type operators we discuss a specific class of regular singular laplace type operators with matrix coefficients their zeta determinants were studied by kirsten loya and park manuscr math 125 95 2008 on the basis of the contour integral method with general boundary conditions at the singularity and dirichlet boundary conditions at the regular boundary we complete the arguments of kirsten loya and park by explicitly verifying that the contour integral method indeed applies in the regular singular setup further we extend the zeta determinant computations to generalized neumann boundary conditions at the regular boundary and apply our results to compute zeta determinants of laplacians on a bounded generalized cone with relative ideal boundary conditions

the bar partial operator on algebraic curves for a singular algebraic curve we show that all closed extensions of bar partial are fredholm and we give a general index formula in particular we prove a modified version of a conjecture due to macpherson

an equivariant index for proper actions i abstract equivariant indices taking values in group theoretic objects have previously been defined in cases where either the group acting or the orbit space of the action is compact in this paper we define an equivariant index without assuming the group or the orbit space to be compact this allows us to generalise an index of deformed dirac operators defined for compact groups by braverman in parts ii and iii of this series we explore some properties and applications of this index

sur la l 2 cohomologie des varietes a courbure negative we give a topological interpretation of the space of l2 harmonic forms on finite volume manifolds with sufficiently pinched negative curvature we give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants the method consists first in comparing l2 cohomology with weighted l2 cohomology thanks to previuos works done by t ohsawa and then in identifying these weighted spaces

higher order analogues of tracy widom distributions via the lax method we study the distribution of the largest eigenvalue in hermitian one matrix models when the spectral density acquires an extra number of k 1 zeros at the edge the distributions are directly expressed through the norms of orthogonal polynomials on a semi infinite interval as an alternative to using fredholm determinants they satisfy nonlinear recurrence relations which we show form a lax pair making contact to the string literature in the early 1990s the technique of pseudo differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the painleve xxxiv hierarchy these are the higher order analogues of the tracy widom distribution using known backlund transformations we show how to simplify earlier equivalent results that are derived from fredholm determinant theory valid for even k in terms of the painleve ii hierarchy

heat kernels and functional determinants on the generalized cone we consider zeta functions and heat kernel expansions on the bounded generalized cone in arbitrary dimensions using an improved calculational technique the specific case of a global monopole is analysed in detail and some restrictions thereby placed on thea 5 2 coefficient the computation of functional determinants is also addressed general formulas are given and known results are incidentally and rapidly reproduced

the numerical analysis of random particle methods applied to vlasov poisson fokker planck kinetic equations we devise and study a random particle method for approximating vlasov poisson fokker planck systems such a proposed scheme takes into account the fact that the trajectories of a particle undergoing brownian motion due to collisions with the medium or background particles can be obtained as the solutions of stochastic differential equations i e the langevin equations these equations are the precise analogs of the deterministic hamiltonian system in the collisionless model the particle approximation in particular simulates the action of viscosity by the use of independent wiener processes brownian motions the error analysis generalizes that of ganguly and victory siam j numer anal 26 1989 pp 249 288 by incorporating into the consistency estimates the sampling errors generated by the random motion of the particles these particular errors are the dominant component of the consistency errors and are estimated by applying bennett s inequality for estimating tail probabilities for standard

mesoscopic eigenvalue density correlations of wigner matrices we investigate to what extent the microscopic wigner gaudin mehta dyson wgmd or sine kernel statistics of random matrix theory remain valid on mesoscopic scales to that end we compute the connected two point spectral correlation function of a wigner matrix at two mesoscopically separated points in the mesoscopic regime density correlations are much weaker than in the microscopic regime our result is an explicit formula for the two point function this formula implies that the wgmd statistics are valid to leading order on all mesoscopic scales that in the real symmetric case there are subleading corrections matching precisely the wgmd statistics while in the complex hermitian case these subleading corrections are absent we also uncover non universal subleading correlations which dominate over the universal ones beyond a certain intermediate mesoscopic scale the proof is based on a hierarchy of schwinger dyson equations for a sufficiently large class of polynomials in the entries of the green function the hierarchy is indexed by a tree whose depth is controlled using stopping rules a key ingredient in the derivation of the stopping rules is a new estimate on the density of states which we prove to have bounded derivatives of all order on all mesoscopic scales

zeta function determinant of the laplace operator on the d dimensional ball we present a direct approach for the calculation of functional determinants of the laplace operator on balls dirichlet and robin boundary conditions are considered using this approach formulas for any value of the dimension d of the ball can be obtained quite easily explicit results are presented here for dimensionsd 2 3 4 5 and 6

the convergence analysis of fully discretized particle methods for solving vlasov poisson systems error estimates for fully discretized particle or simulation methods for multidimensional vlasov poisson systems are provided the treatment requires stability results which generalize to the vlasov poisson context those of j t beale and a majda math comp 39 1982 pp 29 52 and those of c anderson and c greengard siam j numer anal 22 1985 pp 413 440 for studying the convergence of vortex methods with these stability results the total error at each timestep can be studied this approach produces estimates better than those which would be obtained by considering the errors in the semidiscrete approximation siam j numer anal 26 1989 pp 249 288 separate from the errors incurred in the time discretization of the mollified hamiltonian trajectory equations

wavelet based density estimation for noise reduction in plasma simulations using particles for given computational resources the accuracy of plasma simulations using particles is mainly limited by the noise due to limited statistical sampling in the reconstruction of the particle distribution function a method based on wavelet analysis is proposed and tested to reduce this noise the method known as wavelet based density estimation wbde was previously introduced in the statistical literature to estimate probability densities given a finite number of independent measurements its novel application to plasma simulations can be viewed as a natural extension of the finite size particles fsp approach with the advantage of estimating more accurately distribution functions that have localized sharp features the proposed method preserves the moments of the particle distribution function to a good level of accuracy has no constraints on the dimensionality of the system does not require an a priori selection of a global smoothing scale and its able to adapt locally to the smoothness of the density based on the given discrete particle data moreover the computational cost of the denoising stage is of the same order as one time step of a fsp simulation the method is compared with a recently proposed proper orthogonal decomposition based method and it is tested with three particle data sets involving different levels of collisionality and interaction with external and self consistent fields

interacting brownian particles and the wigner law in this paper we study interacting diffusing particles governed by the stochastic differential equationsdx j t n db j t d jon x 1 x n dt j 1 2 n here theb jare independent brownian motions in d and o n x 1 x n n i j v x i x j n i u x 1 the potentialv has a singularity at 0 strong enough to keep the particles apart and the potentialu serves to keep the particles from escaping to infinity our interest is in the behaviour as the number of particles increases without limit which we study through the empirical measure process we prove tightness of these processes in the case ofd 1 v x log x u x x 2 2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure valued process exists this process is deterministic and converges to the wigner law ast some information on the rates of convergence is derived and the case of a cauchy initial distribution is analysed completely

higher order analogues of the tracy widom distribution and the painleve ii hierarchy we study fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points the kernels are natural higher order analogues of the airy kernel and

bulk universality for deformed wigner matrices we consider n n random matrices of the form h w v where w is a real symmetric or complex hermitian wigner matrix and v is a random or deterministic real diagonal matrix whose entries are independent of w we assume subexponential decay for the matrix entries of w and we choose v so that the eigenvalues of w and v are typically of the same order for a large class of diagonal matrices v we show that the local statistics in the bulk of the spectrum are universal in the limit of large n

rigidity and a mesoscopic central limit theorem for dyson brownian motion for general beta β and potentials we study dyson brownian motion with general potential v and for general beta ge 1 for short times t o 1 and under suitable conditions on v we obtain a local law and corresponding rigidity estimates on the particle locations that is with overwhelming probability the particles are close to their classical locations with an almost optimal error estimate under the condition that the density of states of the initial data is bounded below and above down to the scale eta ll t ll 1 we prove a mesoscopic central limit theorem for linear statistics at all scales eta with n 1 ll eta ll t