universality results for the largest eigenvalues of some sample covariance matrix ensembles for sample covariance matrices with i i d entries with sub gaussian tails when both the number of samples and the number of variables become large and the ratio approaches one it is a well known result of soshnikov that the limiting distribution of the largest eigenvalue is same that of gaussian samples in this paper we extend this result to two cases the first case is when the ratio approaches an arbitrary finite value the second case is when the ratio becomes infinite or arbitrarily small

hypothesis testing for automated community detection in networks summary community detection in networks is a key exploratory tool with applications in a diverse set of areas ranging from finding communities in social and biological networks to identifying link farms in the world wide web the problem of finding communities or clusters in a network has received much attention from statistics physics and computer science however most clustering algorithms assume knowledge of the number of clusters k we propose to determine k automatically in a graph generated from a stochastic block model by using a hypothesis test of independent interest our main contribution is twofold first we theoretically establish the limiting distribution of the principal eigenvalue of the suitably centred and scaled adjacency matrix and use that distribution for our test of the hypothesis that a random graph is of erd s renyi noise type secondly we use this test to design a recursive bipartitioning algorithm which naturally uncovers nested community structure using simulations and quantifiable classification tasks on real world networks with ground truth we show that our algorithm outperforms state of the art methods

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

universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices we prove universality at the edge of the spectrum for unitary 2 orthogonal 1 and symplectic 4 ensembles of random matrices in the scaling limit for a class of weights w x e v x where v is a polynomial v x 2mx2m 2m 0 the precise statement of our results is given in theorem 1 1 and corollaries 1 2 and 1 4 below for the same class of weights a proof of universality in the bulk of the spectrum is given in 12 for the unitary ensembles and in 9 for the orthogonal and symplectic ensembles our starting point in the unitary case is 12 and for the orthogonal and symplectic cases we rely on our recent work 9 which in turn depends on the earlier work of widom 46 and tracy and widom 42 as in 9 the uniform plancherel rotach type asymptotics for the orthogonal polynomials found in 12 plays a central role the formulae in 46 express the correlation kernels for 1 4 as a sum of a christoffel darboux cd term as in the case 2 together with a correction term in the bulk scaling limit 9 the correction term is of lower order and does not contribute to the limiting form of the correlation kernel by contrast in the edge scaling limit considered here the cd term and the correction term contribute to the same order this leads to additional technical difficulties over and above 49 2006 wiley periodicals inc

local marchenko pastur law at the hard edge of sample covariance matrices let xn be a n n matrix whose entries are independent identically distributed complex random variables with mean zero and variance 1n we study the asymptotic spectral distribution of the eigenvalues of the covariance matrix xn xn for n we prove that the empirical density of eigenvalues in an interval e e converges to the marchenko pastur law locally on the optimal scale n e logn b and in any interval up to the hard edge logn bn2 e 4 for any 0 as a consequence we show the complete delocalization of the eigenvectors

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

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

finite n fluctuation formulas for random matrices for the gaussian and laguerre random matrix ensembles the probability density function p d f for the linear statistic jn 1 xj x is computed exactly and shown to satisfy a central limit theorem asn for the circular random matrix ensemble the p d f s for the statistics jn 1 j and jn 1 log 2 sin j 2 are calculated exactly by using a constant term identity from the theory of the selberg integral and are also shown to satisfy a central limit theorem asn

the eigenvector moment flow and local quantum unique ergodicity we prove that the distribution of eigenvectors of generalized wigner matrices is universal both in the bulk and at the edge this includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries the proof relies on analyzing the eigenvector flow under the dyson brownian motion the key new ideas are 1 the introduction of the eigenvector moment flow a multi particle random walk in a random environment 2 an effective estimate on the regularity of this flow based on maximum principle and 3 optimal finite speed of propagation holds for the eigenvector moment flow with very high probability

numerical range for random matrices abstract we analyze the numerical range of high dimensional random matrices obtaining limit results and corresponding quantitative estimates in the non limit case for a large class of random matrices their numerical range is shown to converge to a disc in particular numerical range of complex ginibre matrix almost surely converges to the disk of radius 2 since the spectrum of non hermitian random matrices from the ginibre ensemble lives asymptotically in a neighborhood of the unit disk it follows that the outer belt of width 2 1 containing no eigenvalues can be seen as a quantification the non normality of the complex ginibre random matrix we also show that the numerical range of upper triangular gaussian matrices converges to the same disk of radius 2 while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to 2 e

fluctuations of matrix entries of regular functions of wigner matrices we study the fluctuations of the matrix entries of regular functions of wigner random matrices in the limit when the matrix size goes to infinity in the case of the gaussian ensembles goe and gue this problem was considered by a lytova and l pastur j stat phys 134 147 159 2009 our results are valid provided the off diagonal matrix entries have finite fourth moment the diagonal matrix entries have finite second moment and the test functions have four continuous derivatives in a neighborhood of the support of the wigner semicircle law moreover if the marginal distributions satisfy the poincare inequality our results are valid for lipschitz continuous test functions

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

global attractors for nonlinear parabolic equations with nonstandard growth and irregular data abstract we investigate the large time behavior of solutions to the following nonlinear parabolic equations u t div u p x 2 u f x u g in r u 0 on r u x 0 u 0 x in where u 0 g l 1 we first provide the existence and uniqueness of an entropy solution for the problem then through some delicate analysis we establish some regularity results on the solution by which we prove the existence of a global attractor for the solution semigroup

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

on superconvergence of sums of free random variables this paper derives sufficient conditions for superconvergence of sums of bounded free random variables and provides an estimate for the rate of superconvergence

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

average density of states for hermitian wigner matrices abstract we consider ensembles of n n hermitian wigner matrices whose entries are up to the symmetry constraints independent and identically distributed random variables assuming sufficient regularity for the probability density function of the entries we show that the expectation of the density of states on arbitrarily small intervals converges to the semicircle law as n tends to infinity

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

random tensor theory extending random matrix theory to mixtures of random product states we consider a problem in random matrix theory that is inspired by quantum information theory determining the largest eigenvalue of a sum of p random product states in mathbb c d otimes k where k and p d k are fixed while d when k 1 the marcenko pastur law determines up to small corrections not only the largest eigenvalue 1 sqrt p d k 2 but the smallest eigenvalue min 0 1 sqrt p d k 2 and the spectral density in between we use the method of moments to show that for k 1 the largest eigenvalue is still approximately 1 sqrt p d k 2 and the spectral density approaches that of the marcenko pastur law generalizing the random matrix theory result to the random tensor case our bound on the largest eigenvalue has implications both for sampling from a particular heavy tailed distribution and for a recently proposed quantum data hiding and correlation locking scheme due to leung and winter

determinants of airy operators and applications to random matrices the purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of wiener hopf operators the determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum

large gap asymptotics for airy kernel determinants with discontinuities we obtain large gap asymptotics for airy kernel fredholm determinants with any number m of discontinuities these m point determinants are generating functions for the airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles our main result is that the m point determinants can be expressed asymptotically as the product of m 1 point determinants multiplied by an explicit constant pre factor which can be interpreted in terms of the covariance of the counting function of the process

on the relation between orthogonal symplectic and unitary matrix ensembles for the unitary ensembles of n n hermitian matrices associated with a weight function w there is a kernel expressible in terms of the polynomials orthogonal with respect to the weight function which plays an important role for the orthogonal and symplectic ensembles of hermitian matrices there are 2 2 matrix kernels usually constructed using skew orthogonal polynomials which play an analogous role these matrix kernels are determined by their upper left hand entries we derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles we also show that whenever w w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w w general formulas are obtained for these extra terms we do not use skew orthogonal polynomials in the derivations

wigner random matrices with non symmetrically distributed entries we show that the spectral radius of an n n random symmetric matrix with i i d bounded centered but non symmetrically distributed entries is bounded from above by 2 o n 6 11 e where 2 is the variance of the matrix entries and e is an arbitrary small positive number our bound improves the earlier results by z furedi and j komlos 1981 and van vu 2005

on the largest singular values of random matrices with independent cauchy entries we apply the method of determinants to study the distribution of the largest singular values of large m n real rectangular random matrices with independent cauchy entries we show that for a special one parametric class of statistics the properties of the largest singular values rescaled by a factor 1 m2n2 agree in the limit with the statistical properties of the poisson random point process with the intensity 1 x 3 2 and therefore are different from the tracy widom law among other corollaries of our method we show an interesting connection between the mathematical expectations of the determinants of the complex rectangular m n standard wishart ensemble and the real rectangular 2m 2n standard wishart ensemble

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

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

eigenvector distribution of wigner matrices we consider n n hermitian or symmetric random matrices with independent entries the distribution of the i j th matrix element is given by a probability measure ij whose first two moments coincide with those of the corresponding gaussian ensemble we prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding gaussian ensemble for eigenvectors associated with bulk eigenvalues the same conclusion holds provided the first four moments of the distribution ij coincide with those of the corresponding gaussian ensemble more generally we prove that the joint eigenvector eigenvalue distributions near the spectral edge of two generalized wigner ensembles agree provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate if in addition the first four moments match then this result holds also in the bulk

a note on the eigenvalue density of random matrices the distribution of eigenvalues of n n random matrices in the limit n is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges this fact is a consequence of a more general theorem proven here in the statistical mechanics of unstable interactions our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems

on universality of local edge regime for the deformed gaussian unitary ensemble we consider the deformed gaussian ensemble h n h n 0 m n in which h n 0 is a hermitian matrix possibly random and m n is the gaussian unitary random matrix gue independent of h n 0 assuming that the normalized counting measure of h n 0 converges weakly in probability if random to a non random measure n 0 with a bounded support and assuming some conditions on the convergence rate we prove the universality of the local eigenvalue statistics near the edge of the limiting spectrum of h n

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

on the distributions of the lengths of the longest monotone subsequences in random words we consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length n from an alphabet of k letters in the limit as k these become the corresponding distributions for permutations on n letters we find toeplitz determinant representations for the exponential generating functions on n of these distribution functions and show that they are expressible in terms of solutions of painleve v equations we show further that in the weakly increasing case the generating unction gives the distribution of the smallest eigenvalue in the k k laguerre random matrix ensemble and that the distribution itself has after centering and normalizing an n limit which is equal to the distribution function for the largest eigenvalue in the gaussian unitary ensemble of k k hermitian matrices of trace zero

universality of the edge distribution of eigenvalues of wigner random matrices with polynomially decaying distributions of entries we consider an ensemble of wigner symmetric random matrices a n a ij i j 1 n with matrix elements a ij being i i d symmetrically distributed random variables open image in new window we assume that open image in new window and that

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

random unitary matrices methods of constructing random matrices typical of circular unitary and circular orthogonal ensembles are presented we generate numerically random unitary matrices and show that the statistical properties of their spectra level spacing distribution number variance and eigenvectors entropy participation ratio eigenvector statistics confer to the predictions of the random matrix theory for both cue and coe

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

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

the spectrum of heavy tailed random matrices let x n be an n n random symmetric matrix with independent equidistributed entries if the law p of the entries has a finite second moment it was shown by wigner 14 that the empirical distribution of the eigenvalues of x n once renormalized by sqrt n converges almost surely and in expectation to the so called semicircular distribution as n goes to infinity in this paper we study the same question when p is in the domain of attraction of an stable law we prove that if we renormalize the eigenvalues by a constant a n of order n frac 1 alpha the corresponding spectral distribution converges in expectation towards a law mu alpha which only depends on we characterize mu alpha and study some of its properties it is a heavy tailed probability measure which is absolutely continuous with respect to lebesgue measure except possibly on a compact set of capacity zero

on eigenvalues of the sum of two random projections we study the behavior of eigenvalues of matrix p n q n where p n and q n are two n by n random orthogonal projections we relate the joint eigenvalue distribution of this matrix to the jacobi matrix ensemble and establish the universal behavior of eigenvalues for large n the limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the bessel or the airy kernel at the edge depending on parameters we also study an exceptional case when the local behavior of eigenvalues of p n q n is not universal in the usual sense

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn

the drazin inverses of products and differences of orthogonal projections in this note the drazin inverses of products and differences of orthogonal projections on a hilbert space are established

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

the largest eigenvalues of sample covariance matrices for a spiked population diagonal case we consider large complex random sample covariance matrices obtained from spiked populations that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to 1 we investigate the limiting behavior of the largest eigenvalues when the population and the sample sizes both become large under some conditions on moments of the sample distribution we prove that the asymptotic fluctuations of the largest eigenvalues are the same as for a complex gaussian sample with the same true covariance the real setting is also considered

symmetric pseudo random matrices we consider the problem of generating symmetric pseudo random sign 1 matrices based on the similarity of their spectra to wigner s semicircular law using binary m sequences golomb sequences of lengths n 2 m 1 we give a simple explicit construction of circulant n times n sign matrices and show that their spectra converge to the semicircular law when n grows the kolmogorov complexity of the proposed matrices equals to that of golomb sequences and is at most 2 mathrm log 2 n bits

on asymptotic expansions and scales of spectral universality in band random matrix ensembles we consider real random symmetric n n matrices h of the band type form with characteristic length b the matrix entries are independent gaussian random variables and have the variance proportional to where u t vanishes at infinity we study the resolvent in the limit and obtain the explicit expression for the leading term of the first correlation function of the normalized trace

a limit theorem at the edge of a non hermitian random matrix ensemble the study of the edge behaviour in the classical ensembles of gaussian hermitian matrices has led to the celebrated distributions of tracy widom here we take up a similar line of inquiry in the non hermitian setting we focus on the family of n n random matrices with all entries independent and distributed as complex gaussian of mean zero and variance 1 n this is a fundamental non hermitian ensemble for which the eigenvalue density is known using this density our main result is a limit law for the scaled spectral radius as n as a corollary we get the analogous statement for the case where the complex gaussians are replaced by quaternion gaussians

asymptotic properties of large random matrices with independent entries we study the normalized trace gn z n 1 tr h zi 1 of the resolvent of n n real symmetric matrices h 1 jk wjk n j k 1n assuming that their entries are independent but not necessarily identically distributed random variables we develop a rigorous method of asymptotic analysis of moments of gn z for iz 0 where 0 is determined by the second moment of wjk by using this method we find the asymptotic form of the expectation e gn z and of the connected correlator e gn z1 gn z2 e gn z1 e gn z2 we also prove that the centralized trace ngn z e ngn z has the gaussian distribution in the limit n based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles

scaled limit and rate of convergence for the largest eigenvalue from the generalized cauchy random matrix ensemble in this paper we are interested in the asymptotic properties for the largest eigenvalue of the hermitian random matrix ensemble called the generalized cauchy ensemble gcye whose eigenvalues pdf is given by textrm const cdot prod 1 leq j k leq n x j x k 2 prod j 1 n 1 ix j s n 1 ix j overline s n dx j where s is a complex number such that s 1 2 and where n is the size of the matrix ensemble using results by borodin and olshanski commun math phys 223 1 87 123 2001 we first prove that for this ensemble the law of the largest eigenvalue divided by n converges to some probability distribution for all s such that s 1 2 using results by forrester and witte nagoya math j 174 29 114 2002 on the distribution of the largest eigenvalue for fixed n we also express the limiting probability distribution in terms of some non linear second order differential equation eventually we show that the convergence of the probability distribution function of the re scaled largest eigenvalue to the limiting one is at least of order 1 n

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices