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

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

local kesten mckay law for random regular graphs we study the adjacency matrices of random d regular graphs with large but fixed degree d in the bulk of the spectrum 2 sqrt d 1 varepsilon 2 sqrt d 1 varepsilon down to the optimal spectral scale we prove that the green s functions can be approximated by those of certain infinite tree like few cycles graphs that depend only on the local structure of the original graphs this result implies that the kesten mckay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors our method is based on estimating the green s function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs

generic results in classes of ultradifferentiable functions abstract let e be a denjoy carleman class of ultradifferentiable functions of beurling type on the real line that strictly contains another class f of roumieu type we show that the set s of functions in e that are nowhere in the class f is large in the topological sense it is residual in the measure theoretic sense it is prevalent and that s 0 contains an infinite dimensional linear subspace it is lineable consequences for the gevrey classes are given similar results are also obtained for classes of ultradifferentiable functions defined imposing conditions on the fourier laplace transform of the function

extremal eigenvalues and eigenvectors of deformed wigner matrices we consider random matrices of the form h w lambda v lambda in mathbb r where w is a real symmetric or complex hermitian wigner matrix of size n and v is a real bounded diagonal random matrix of size n with i i d entries that are independent of w we assume subexponential decay of the distribution of the matrix entries of w and we choose lambda sim 1 so that the eigenvalues of w and lambda v are typically of the same order further we assume that the density of the entries of v is supported on a single interval and is convex near the edges of its support in this paper we prove that there is lambda in mathbb r such that the largest eigenvalues of h are in the limit of large n determined by the order statistics of v for lambda lambda in particular the largest eigenvalue of h has a weibull distribution in the limit n rightarrow infty if lambda lambda moreover for n sufficiently large we show that the eigenvectors associated to the largest eigenvalues are partially localized for lambda lambda while they are completely delocalized for lambda lambda similar results hold for the lowest eigenvalues

how far is the borel map from being surjective in quasianalytic ultradifferentiable classes abstract the borel map j takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0 in the literature it is well known that the restriction of j to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space in this paper we are interested in studying how large the image of j is and we investigate the size and the structure of this image by using different approaches baire residuality prevalence and lineability we give an answer to this question in the very general setting of quasianalytic ultradifferentiable classes defined by weight matrices which contains as particular cases the classes defined by a single weight sequence or by a weight function

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

on the number of closed geodesics on projective spaces 1 let pro k be the m dimensional projective space over the complex numbers k 2 the quaternions k 4 or the cayley plane k 8 m 2 endowed with the standard riemannian metric and m be a compact riemannian manifold and f pm k m a differentiable homotopy equivalence a circle on pro k is mapped under f onto a prime closed curve of length 2 hingston proved in 5 w 5 that under these conditions there are g k m distinct closed geodesics without self intersections and of length g k m in most cases it is g l k m g k m s k 2 against that on the standard sphere s the cup length of the space of unoriented parametrized great circles is always smaller than the cup length of the space of unparametrized great circles for the case of the sphere see 8 ch 3 6

delocalization for a class of random block band matrices we consider n times n hermitian random matrices h consisting of blocks of size m ge n 6 7 the matrix elements are i i d within the blocks close to a gaussian in the four moment matching sense but their distribution varies from block to block to form a block band structure with an essential band width m we show that the entries of the green s function g z h z 1 satisfy the local semicircle law with spectral parameter z e mathbf i eta down to the real axis for any eta gg n 1 using a combination of the supersymmetry method inspired by shcherbina j stat phys 155 3 466 499 2014 and the green s function comparison strategy previous estimates were valid only for eta gg m 1 the new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized

prevalence a translation invariant almost every on infinite dimensional spaces we present a measure theoretic condition for a property to hold almost everywhere on an infinite dimensional vector space with particularemphasis on function spaces such as c k and l p like the concept of lebesgue almost every on an infinite dimensional spaces our notion of prevalence is translation invariant instead of using a specific measure on the entire space we define prevalence in terms of the class of all probability measures with compact support prevalence is a more appropriate condition than the topological concepts of open and dense or generic when one desires a probabilistic result on the likelihood of a given property on a function space

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

sparse general wigner type matrices local law and eigenvector delocalization we prove a local law and eigenvector delocalization for general wigner type matrices our methods allow us to get the best possible interval length and optimal eigenvector delocalization in the dense case and the first results of such kind for the sparse case down to p g n log nn with g n we specialize our results to the case of the stochastic block model and we also obtain a local law for the case when the number of classes is unbounded

local semicircle law for random regular graphs we consider random d regular graphs on n vertices with degree d at least log n 4 we prove that the green s function of the adjacency matrix and the stieltjes transform of its empirical spectral measure are well approximated by wigner s semicircle law down to the optimal scale given by the typical eigenvalue spacing up to a logarithmic correction aside from well known consequences for the local eigenvalue distribution this result implies the complete isotropic delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity 2017 wiley periodicals inc

local law and tracy widom limit for sparse random matrices we consider spectral properties and the edge universality of sparse random matrices the class of random matrices that includes the adjacency matrices of the erd s renyi graph model g n p we prove a local law for the eigenvalue density up to the spectral edges under a suitable condition on the sparsity we also prove that the rescaled extremal eigenvalues exhibit goe tracy widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included for the adjacency matrix of the erd s renyi graph this establishes the tracy widom fluctuations of the second largest eigenvalue when p is much larger than n 2 3 with a deterministic shift of order np 1

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

bulk universality of sparse random matrices we consider the adjacency matrix of the ensemble of erd s renyi random graphs which consists of graphs on n vertices in which each edge occurs independently with probability p we prove that in the regime pn 1 these matrices exhibit bulk universality in the sense that both the averaged n point correlation functions and distribution of a single eigenvalue gap coincide with those of the goe our methods extend to a class of random matrices which includes sparse ensembles whose entries have different variances

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

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

delocalization at small energy for heavy tailed random matrices we prove that the eigenvectors associated to small enough eigenvalues of an heavy tailed symmetric random matrix are delocalized with probability tending to one as the size of the matrix grows to infinity the delocalization is measured thanks to a simple criterion related to the inverse participation ratio which computes an average ratio of l 4 and l 2 norms of vectors in contrast as a consequence of a previous result for random matrices with sufficiently heavy tails the eigenvectors associated to large enough eigenvalues are localized according to the same criterion the proof is based on a new analysis of the fixed point equation satisfied asymptotically by the law of a diagonal entry of the resolvent of this matrix

bulk eigenvalue statistics for random regular graphs ak was partly supported by swiss national science foundation grant 144662 hty was partly supported by nsf grant dms 1307444 and a simons investigator fellowship

the spectral edge of some random band matrices we study the asymptotic distribution of the eigenvalues of random hermitian periodic band matrices focusing on the spectral edges the eigenvalues close to the edges converge in distribution to the airy point process if and only if the band is sufficiently wide w n 5 6 otherwise a different limiting distribution appears

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e 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 under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

local spectral statistics of the addition of random matrices we consider the local statistics of h v x v u y u where v and u are independent haar distributed unitary matrices and x and y are deterministic real diagonal matrices in the bulk we prove that the gap statistics and correlation functions coincide with the gue in the limit when the matrix size n rightarrow infty under mild assumptions on x and certain rigidity assumptions on y the latter being an assumption on the convergence of the eigenvalues of y to the quantiles of its limiting spectral measure which we assume to have a density our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to dyson brownian motion our method also applies to the case when v and u are drawn from the orthogonal group our proof relies on the local law for h proved in bao et al commun math phys 349 3 947 990 2017 j funct anal 271 3 672 719 2016 adv math 319 251 291 2017 as well as the dbm convergence results of landon and yau commun math phys 355 3 949 1000 2017 and landon et al adv math 346 1137 1332 2019

anisotropic local laws for random matrices we develop a new method for deriving local laws for a large class of random matrices it is applicable to many matrix models built from sums and products of deterministic or independent random matrices in particular it may be used to obtain local laws for matrix ensembles that are anisotropic in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity for definiteness we present the method for sample covariance matrices of the form open image in new window where t is deterministic and x is random with independent entries we prove that with high probability the resolvent of q is close to a deterministic matrix with an optimal error bound and down to optimal spectral scales as an application we prove the edge universality of q by establishing the tracy widom airy statistics of the eigenvalues of q near the soft edges this result applies in the single cut and multi cut cases further applications include the distribution of the eigenvectors and an analysis of the outliers and bbp type phase transitions in finite rank deformations they will appear elsewhere we also apply our method to wigner matrices whose entries have arbitrary expectation i e we consider w a where w is a wigner matrix and a a hermitian deterministic matrix we prove the anisotropic local law for w a and use it to establish edge universality

bulk universality and related properties of hermitian matrix models we give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models assuming that the potential that determines the model is globally c 2 and locally c 3 function see theorem 3 1 the proof as our previous proof in pastur and shcherbina in j stat phys 86 109 147 1997 is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials rather we obtain the sin kernel as a unique solution of a certain non linear integro differential equation that follows from the determinant formulas for the correlation functions of the model we also give a simplified and strengthened version of paper boutet de monvel et al in j stat phys 79 585 611 1995 on the existence and properties of the limiting normalized counting measure of eigenvalues we use these results in the proof of universality and we believe that they are of independent interest

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