central limit theorem for linear eigenvalue statistics of random matrices with independent entries we consider n x n real symmetric and hermitian wigner random matrices n 1 2 w with independent modulo symmetry condition entries and the null sample covariance matrices n 1 x x with independent entries of m x n matrix x assuming first that the 4th cumulant excess 4 of entries of w and x is zero and that their 4th moments satisfy a lindeberg type condition we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem clt as n m m n c 0 oo with the same variance as for gaussian matrices if the test functions of statistics are smooth enough essentially of the class c 5 this is done by using a simple interpolation trick from the known results for the gaussian matrices and the integration by parts presented in the form of certain differentiation formulas then by using a more elaborated version of the techniques we prove the clt in the case of nonzero excess of entries again for essentially 5 test function here the variance of statistics contains an additional term proportional to 4 the proofs of all limit theorems follow essentially the same scheme

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

spectra of sparse random matrices we compute the spectral density for ensembles of sparse symmetric random matrices using replica our formulation of the replica symmetric ansatz shares the symmetries of that suggested in a seminal paper by rodgers and bray symmetry with respect to permutation of replica and rotation symmetry in the space of replica but uses a different representation in terms of superpositions of gaussians it gives rise to a pair of integral equations which can be solved by a stochastic population dynamics algorithm remarkably our representation allows us to identify pure point contributions to the spectral density related to the existence of normalizable eigenstates our approach is not restricted to matrices defined on graphs with poissonian degree distribution matrices defined on regular random graphs or on scale free graphs are easily handled we also look at matrices with row constraints such as discrete graph laplacians our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordinations and an example of such an unfolding is presented our results are well corroborated by numerical diagonalization studies of large finite random matrices

universality of level correlation function of sparse random matrices the statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the hubbard stratonovich transformation when used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick the model turns out to bear a close resemblance to the anderson model on the bethe lattice it possesses a delocalization transition that occurs with an increase in the mean connectivity parameter in the delocalized phase the level level correlation function proves to have a universal dyson form with the full density of states replaced by the contribution from the infinite cluster

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

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

clt for linear spectral statistics of large dimensional sample covariance matrices let bn 1 n t1 2nxnx nt1 2n where xn xij is n n with i i d complex standardized entries having finite fourth moment and t1 2n is a hermitian square root of the nonnegative definite hermitian matrix tn the limiting behavior as n with n n approaching a positive constant of functionals of the eigenvalues of bn where each is given equal weight is studied due to the limiting behavior of the empirical spectral distribution of bn it is known that these linear spectral statistics converges a s to a nonrandom quantity this paper shows their rate of convergence to be 1 n by proving after proper scaling that they form a tight sequence moreover if exppx211 0 and expp x11 4 2 or if x11 and tn are real and exppx411 3 they are shown to have gaussian limits

eigenvalues outside the bulk of inhomogeneous erdős renyi random graphs in this article an inhomogeneous erd s renyi random graph on 1 ldots n is considered where an edge is placed between vertices i and j with probability varepsilon n f i n j n for i le j the choice being made independently for each pair the integral operator i f associated with the bounded function f is assumed to be symmetric non negative definite and of finite rank k we study the edge of the spectrum of the adjacency matrix of such an inhomogeneous erd s renyi random graph under the assumption that n varepsilon n rightarrow infty sufficiently fast although the bulk of the spectrum of the adjacency matrix scaled by sqrt n varepsilon n is compactly supported the kth largest eigenvalue goes to infinity it turns out that the largest eigenvalue after appropriate scaling and centering converges to a gaussian law if the largest eigenvalue of i f has multiplicity 1 if i f has k distinct non zero eigenvalues then the joint distribution of the k largest eigenvalues converge jointly to a multivariate gaussian law the first order behaviour of the eigenvectors is derived as a byproduct of the above results the results complement the homogeneous case derived by 18

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

regular sparse crossbar concentrators a bipartite concentrator is a single stage sparse crossbar switching device that can connect any m of its n spl ges m inputs to its m outputs possibly without the ability to distinguish their order fat and slim crossbars were introduced recently to show that bipartite concentrators can be constructed with a minimum number of crosspoints for any number of inputs and outputs we generalize these graphs to obtain bipartite concentrators with nearly a fixed fanout without altering their n m 1 m crosspoint complexity we also present an o log n time algorithm to route arbitrary concentration assignments on this new family of fat and slim crossbars

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

a numerical study of sparse random matrices a numerical study is presented for the eigensolution statistics of largen n real and symmetric sparse random matrices as a function of the mean numberp of nonzero elements per row the model shows classical percolation and quantum localization transitions atp c 1 andp q 1 respectively in the rigid limitp n we demonstrate that the averaged density of states follows the wigner semicircle law and the corresponding nearest energy level spacing distribution functionp s obeys the wigner surmise in the very sparse matrix limitp n withp p q a singularity e 1 e is found as e 0 and exponential tails develop in the high e regions but thep s distribution remains consistent with level repulsion the localization properties of the model are examined by studying both the eigenvector amplitude and the density fluctuations the valuep q 1 4 is roughly estimated in agreement with previous studies of the anderson transition in dilute bethe lattices

inferring hidden states in langevin dynamics on large networks average case performance we present average performance results for dynamical inference problems in large networks where a set of nodes is hidden while the time trajectories of the others are observed examples of this scenario can occur in signal transduction and gene regulation networks we focus on the linear stochastic dynamics of continuous variables interacting via random gaussian couplings of generic symmetry we analyze the inference error given by the variance of the posterior distribution over hidden paths in the thermodynamic limit and as a function of the system parameters and the ratio ensuremath alpha between the number of hidden and observed nodes by applying kalman filter recursions we find that the posterior dynamics is governed by an effective drift that incorporates the effect of the observations we present two approaches for characterizing the posterior variance that allow us to tackle respectively equilibrium and nonequilibrium dynamics the first appeals to random matrix theory and reveals average spectral properties of the inference error and typical posterior relaxation times the second is based on dynamical functionals and yields the inference error as the solution of an algebraic equation

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

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

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 density of states of sparse random matrices the supersymmetric method of calculation of the density of states of a sparse random matrix is shown to be absolutely equivalent to the replica trick a functional generalization of the hubbard stratonovich transformation was used in the course of the solution

diluted banded random matrices scaling behavior of eigenfunction and spectral properties we demonstrate that the normalized localization length of the eigenfunctions of diluted sparse banded random matrices follows the scaling law the scaling parameter of the model is defined as where is the average number of non zero elements per matrix row n is the matrix size and additionally we show that also scales the spectral properties of the model up to certain sparsity characterized by the spacing distribution of eigenvalues

central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs diluted regime we study the linear eigenvalue statistics of large random graphs in the regimes when the mean number of edges for each vertex tends to infinity we prove that for a rather wide class of test functions the fluctuations of linear eigenvalue statistics converges in distribution to a gaussian random variable with zero mean and variance which coincides with non gaussian part of the wigner ensemble variance

universality for the largest eigenvalue of sample covariance matrices with general population this paper is aimed at deriving the universality of the largest eigenvalue of a class of high dimensional real or complex sample covariance matrices of the form mathcal w n sigma 1 2 xx sigma 1 2 here x x ij m n is an m times n random matrix with independent entries x ij 1 leq i leq m 1 leq j leq n such that mathbb e x ij 0 mathbb e x ij 2 1 n on dimensionality we assume that m m n and n m rightarrow d in 0 infty as n rightarrow infty for a class of general deterministic positive definite m times m matrices sigma under some additional assumptions on the distribution of x ij s we show that the limiting behavior of the largest eigenvalue of mathcal w n is universal via pursuing a green function comparison strategy raised in probab theory related fields154 2012 341 407 adv math 229 2012 1435 1515 by erd s yau and yin for wigner matrices and extended by pillai and yin ann appl probab 24 2014 935 1001 to sample covariance matrices in the null case sigma i consequently in the standard complex case mathbb e x ij 2 0 combing this universality property and the results known for gaussian matrices obtained by el karoui in ann probab 35 2007 663 714 nonsingular case and onatski in ann appl probab 18 2008 470 490 singular case we show that after an appropriate normalization the largest eigenvalue of mathcal w n converges weakly to the type 2 tracy widom distribution mathrm tw 2 moreover in the real case we show that when sigma is spiked with a fixed number of subcritical spikes the type 1 tracy widom limit mathrm tw 1 holds for the normalized largest eigenvalue of mathcal w n which extends a result of feral and peche in j math phys 50 2009 073302 to the scenario of nondiagonal sigma and more generally distributed x in summary we establish the tracy widom type universality for the largest eigenvalue of generally distributed sample covariance matrices under quite light assumptions on sigma applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed

random matrices tail bounds for gaps between eigenvalues gaps or spacings between consecutive eigenvalues are a central topic in random matrix theory the goal of this paper is to study the tail distribution of these gaps in various random matrix models we give the first repulsion bound for random matrices with discrete entries and the first super polynomial bound on the probability that a random graph has simple spectrum along with several applications

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

gaussian fluctuations for linear spectral statistics of wigner beta ensembles abstract as an important topic in mathematical physics and statistics random matrices theory has found uses in many aspects of modern physics and multivariate analysis this paper is to investigate the gaussian fluctuations for linear spectral statistics lss of wigner beta ensembles we establish a central limit theorem clt for lss of wigner quaternion matrices

spectral statistics of bernoulli matrix ensembles a random walk approach i we investigate the eigenvalue statistics of random bernoulli matrices where the matrix elements are chosen independently from a binary set with equal probability this is achieved by initiating a discrete random walk process over the space of matrices and analysing the induced random motion of the eigenvalues an approach which is similar to dyson s brownian motion model but with important modifications in particular we show our process is described by a fokker planck equation up to an error margin which vanishes in the limit of large matrix dimension the stationary solution of which corresponds to the joint probability density function of certain well known fixed trace gaussian ensembles

random incidence matrices moments of the spectral density we study numerically and analytically the spectrum of incidence matrices of random labeled graphs on n vertices any pair of vertices is connected by an edge with probability p we give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in n and p for large n and fixed p the spectrum contains a large eigenvalue at np and a semicircle of small eigenvalues for large n and fixed average connectivity pn dilute or sparse random matrices limit we show that the spectrum always contains a discrete component an anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed we develop recursion relations to compute the moments as explicit polynomials in pn their growth is slow enough so that they determine the spectrum the extension of our methods to the laplacian matrix is given in appendix

density of states of sparse random matrices a supersymmetric formalism is used to derive a set of equations giving the density of states of any real symmetric sparse random matrix as a function of the distribution of non zero elements and the mean number of non zero elements per row p in the matrix where the non zero elements take the values or 1 with equal probability the equations are solved as p to infinity recovering results obtained previously with the replica method as e to 0 the density of states rho e behaves as 1 e ln e 2 the more general case where or 1 occur with unequal probabilities is also considered

the isotropic semicircle law and deformation of wigner matrices we analyze the spectrum of additive finite rank deformations of n n wigner matrices h the spectrum of the deformed matrix undergoes a transition associated with the creation or annihilation of an outlier when an eigenvalue di of the deformation crosses a critical value 1 this transition happens on the scale we allow the eigenvalues di of the deformation to depend on n under the condition we make no assumptions on the eigenvectors of the deformation in the limit n we identify the law of the outliers and prove that the nonoutliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a gaussian matrix ensemble a key ingredient in our proof is the isotropic local semicircle law which establishes optimal high probability bounds on where m z is the stieltjes transform of wigner s semicircle law and v w are arbitrary deterministic vectors 2013 wiley periodicals inc

the laplacian on a random one dimensional lattice the authors study the spectrum of minus the laplacian on a random one dimensional lattice it extends monotonically throughout the positive real axis with a continuum limit behaviour for small values the authors find as expected localisation effects throughout the whole range they use a variety of techniques which are shown to be consistent and in agreement with a monte carlo simulation they include small and large disorder expansions direct integral equations for probability distributions as well as the replica method the latter is also used to investigate green functions

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

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

on fluctuations of eigenvalues of random band matrices we consider the fluctuations of linear eigenvalue statistics of random band n times n matrices whose entries have the form mathcal m ij b 1 2 u 1 2 i j b tilde w ij with i i d tilde w ij possessing the 4 varepsilon th moment where the function u has a finite support c c so that m has only 2c b 1 nonzero diagonals the parameter b called the bandwidth is assumed to grow with n in a way such that b n rightarrow 0 without any additional assumptions on the growth of b we prove clt for linear eigenvalue statistics for a rather wide class of test functions thus we improve and generalize the results of the previous papers jana et al arxiv 1412 2445 li et al random matrices 2 04 2013 where clt was proven under the assumption n b n 1 2 moreover we develop a method which allows to prove automatically the clt for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory deformed wigner and sample covariance matrices sparse matrices diluted random matrices matrices with heavy tales etc

bulk universality for generalized wigner matrices with few moments in this paper we consider n times n real generalized wigner matrices whose entries are only assumed to have finite 2 varepsilon th moment for some fixed but arbitrarily small varepsilon 0 we show that the stieltjes transforms m n z of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale when eta mathfrak i z is almost of order n 1 as a consequence we establish bulk universality for local spectral statistics of these matrices at fixed energy levels both in terms of eigenvalue gap distributions and correlation functions meaning that these statistics converge to those of the gaussian orthogonal ensemble in the large n limit

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

universality for general wigner type matrices we consider the local eigenvalue distribution of large self adjoint n times n random matrices mathbf h mathbf h with centered independent entries in contrast to previous works the matrix of variances s i j mathbbm e h i j 2 is not assumed to be stochastic hence the density of states is not the wigner semicircle law its possible shapes are described in the companion paper ajanki et al in quadratic vector equations on the complex upper half plane arxiv 1506 05095 we show that as n grows the resolvent mathbf g z mathbf h z 1 converges to a diagonal matrix mathrm diag mathbf m z where mathbf m z m 1 z dots m n z solves the vector equation 1 m i z z sum j s i j m j z that has been analyzed in ajanki et al quadratic vector equations on the complex upper half plane arxiv 1506 05095 we prove a local law down to the smallest spectral resolution scale and bulk universality for both real symmetric and complex hermitian symmetry classes

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

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

determination of point group harmonics for arbitrary j by a projection method ii icosahedral group quantization along an axis of order 5 the method described in the previous paper to obtain cubic harmonics is applied to the similar problem for the icosahedral group the projection operators are expressed in terms of rotation matrix elements of r 0 and r 0 arctan 2 acting on subspaces invariant under d5

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

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

3 d image reconstruction from averaged fourier transform magnitude by parameter estimation an object model and estimation procedure for three dimensional 3 d reconstruction of objects from measurements of the spherically averaged fourier transform magnitudes is described the motivating application is the 3 d reconstruction of viruses based on solution x ray scattering data the object model includes symmetry positivity and support constraints and has the form of a truncated orthonormal expansion and the parameters are estimated by maximum likelihood methods successful 3 d reconstructions based on synthetic and experimental measurements from cowpea mosaic virus are described

edge universality of correlation matrices xij 0 and e e 2 1 m n m these entries have a sub exponential decay at the tails we will be working in the regime n m dn limn 1 dn 6 0 1 in this paper we prove the edge universality of correlation matrices x x where the rectangular matrix x called the standardized matrix is obtained by normalizing each column of the data matrix e x by its euclidean norm our main result states that asymptotically the k point k 1 correlation functions of the extreme eigenvalues at both edges of the spectrum of the correlation matrix x x converge to those of the gaussian correlation matrix i e tracy widom law and thus in particular the largest and the smallest eigenvalues of x x after appropriate centering and rescaling converge to the tracy widom distribution the asymptotic distribution of extreme eigenvalues of the gaussian correlation matrix has been worked out only recently as a corollary of the main result in this paper we also obtain that the extreme eigenvalues of gaussian correlation matrices are asymptotically distributed according to the tracywidom law the proof is based on the comparison of green functions but the key obstacle to be surmounted is the strong dependence of the entries of the correlation matrix we achieve this via a novel argument which involves comparing the moments of product of the entries of the standardized data matrix to those of the raw data matrix our proof strategy may be extended for proving the edge universality of other random matrix ensembles with dependent entries and hence is of independent interest

critique of the replica trick it is shown that the replica trick fails to give the correct non perturbative result for the two point function s2 of the gaussian unitary ensemble of n n random matrices the failure arises from an incorrect description of the symmetries of the random matrix system in the limit n to infinity the correct description which involves integration over both non compact and compact degrees of freedom is obtained by using the method of superfields some implications for the localisation transition in disordered electronic systems and the theory of the quantised hall effect are suggested

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

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

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 central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

the tracy widom law for some sparse random matrices consider the random matrix obtained from the adjacency matrix of a random d regular graph by multiplying every entry by a random sign the largest eigenvalue converges after proper scaling to the tracy widom distribution

spectral radii of truncated circular unitary matrices abstract consider a truncated circular unitary matrix which is a p n by p n submatrix of an n by n circular unitary matrix by deleting the last n p n columns and rows jiang and qi 11 proved that the maximum absolute value of the eigenvalues known as spectral radius of the truncated matrix after properly normalized converges in distribution to the gumbel distribution if p n n is bounded away from 0 and 1 in this paper we investigate the limiting distribution of the spectral radius under one of the following four conditions 1 p n and p n n 0 as n 2 n p n n 0 and n p n log n 3 as n 3 n p n and n p n log n 0 as n and 4 n p n k 1 is a fixed integer we prove that the spectral radius converges in distribution to the gumbel distribution under the first three conditions and to a reversed weibull distribution under the fourth condition

linear processes in high dimensions phase space and critical properties in this work we investigate the generic properties of a stochastic linear model in the regime of high dimensionality we consider in particular the vector autoregressive var model and the multivariate hawkes process we analyze both deterministic and random versions of these models showing the existence of a stable phase and an unstable phase we find that along the transition region separating the two regimes the correlations of the process decay slowly and we characterize the conditions under which these slow correlations are expected to become power laws we check our findings with numerical simulations showing remarkable agreement with our predictions we finally argue that real systems with a strong degree of self interaction are naturally characterized by this type of slow relaxation of the correlations

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

central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs we consider the adjacency matrix a of a large random graph and study fluctuations of the function fn z u 1 n k 1nexp ugkk z with g z z ia 1 we prove that the moments of fluctuations normalized by n 1 2 in the limit n satisfy the wick relations for the gaussian random variables this allows us to prove central limit theorem for tr g z and then extend the result on the linear eigenvalue statistics tr a of any function r r which increases together with its first two derivatives at infinity not faster than an exponential

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

spectral properties of random reactance networks and random matrix pencils our goal is to study the statistical properies of dielectric resonances which are poles of conductance of a large random lc network such poles are a particular example of eigenvalues n of matrix pencils h w with w being a positive definite matrix and h a random real symmetric one we first consider spectra of the matrix pencils with independent identically distributed entries of h then we concentrate on an infinite range full connectivity version of a random lc network in all cases we calculate the mean eigenvalue density and the two point correlation function in the framework of efetov s supersymmetry approach fluctuations in spectra turn out to be the same as those provided by the wigner dyson theory of usual random matrices

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

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

graph bipartitioning and the bethe spin glass the problem of the equipartitioning of a random graph of fixed finite valence is studied by comparison with a ferromagnetic bethe lattice with random boundary conditions the simplicity of recursion relations for effective fields due to descendents on bethe lattices provides simple approximations for the optimal cost in quite good agreement with simulations

a goodness of fit test for stochastic block models the stochastic block model is a popular tool for studying community structures in network data we develop a goodness of fit test for the stochastic block model the test statistic is based on the largest singular value of a residual matrix obtained by subtracting the estimated block mean effect from the adjacency matrix asymptotic null distribution is obtained using recent advances in random matrix theory the test is proved to have full power against alternative models with finer structures these results naturally lead to a consistent sequential testing estimate of the number of communities

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

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

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

on a labeling for point group harmonics ii icosahedral group the expressions for the 2 5 6 8 and 9 representations are given in terms of those of 1 representations for neighboring angular momenta the coefficients of the 4 representations are expressed in terms of those of 7 representations therefore with an arbitrary choice of orthonormal sets of 1 3 and 7 representations orthonormal sets of other kinds of representations are well defined and can be labeled with the labels of parent representations all clebsch gordan coefficients are expressed in terms of those between parent representations and a few others tables of all nondegenerate 1 3 and 7 representations are given for the axes of quantization of order 5 with some degenerate 3 and 7 representations which are also given any representation of integer or half integer angular momentum up to 27 can be obtained using some usual clebsch gordan coefficients of su 2

stability of the matrix dyson equation and random matrices with correlations we consider real symmetric or complex hermitian random matrices with correlated entries we prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum the correlations have fast decay but are otherwise of general form the key novelty is the detailed stability analysis of the corresponding matrix valued dyson equation whose solution is the deterministic limit of the resolvent

lower bounds on crosspoints in concentrators lower bounds on the required number of crosspoints in concentrators a class of interconnection networks are given the lower bounds are obtained from a straightforward necessary condition on the number of crosspoints in sparse crossbar full capacity concentrators because this condition must be satisfied by all full capacity concentrators embedded in more general concentrators the general necessary condition is established several sparse crossbar designs that contain the minimum number of crosspoints are presented an extension of the results to a more general class of interconnection networks is described

delocalization of eigenvectors of random matrices with independent entries author s rudelson mark vershynin roman abstract we prove that an n by n random matrix g with independent entries is completely delocalized suppose the entries of g have zero means variances uniformly bounded below and a uniform tail decay of exponential type then with high probability all unit eigenvectors of g have all coordinates of magnitude o n 1 2 modulo logarithmic corrections this comes a consequence of a new geometric approach to delocalization for random matrices

bounds for the stieltjes transform and the density of states of wigner matrices we consider ensembles of wigner matrices whose entries are up to the symmetry constraints independent and identically distributed random variables we show the convergence of the stieltjes transform towards the stieltjes transform of the semicircle law on optimal scales and with the optimal rate our bounds improve previous results in particular from erd s et al adv math 229 3 1435 1515 2012 electron j probab 18 59 1 58 2013 by removing the logarithmic corrections as applications we establish the convergence of the eigenvalue counting functions with the rate log n n and the rigidity of the eigenvalues of wigner matrices on the same scale these bounds improve the results of erd s et al adv math 229 3 1435 1515 2012 electron j probab 18 59 1 58 2013 gotze and tikhomirov 2013

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

localization and delocalization of eigenvectors for heavy tailed random matrices consider an n times n hermitian random matrix with above the diagonal independent entries with alpha stable symmetric distribution and 0 2 we give vanishing bounds on the l p norm of the eigenvectors normalized to have unit l 2 norm on the contrary when 0 alpha 2 3 we prove that these eigenvectors are localized

eigenvalue distribution of large weighted random graphs we study eigenvalue distribution of the adjacency matrix a n p of weighted random graphs n p we assume that the graphs have n vertices and the average number of edges attached to one vertex is p to each edge of the graph eij we assign a weight given by a random variable aij with zero mathematical expectation and all moments finite in the first part of the paper we consider the moments of normalized eigenvalue counting function n p of a n p assuming all moments of a finite we obtain recurrent relations that determine the moments of the limiting measure p limn n p the method developed is applied to the laplace operator closely related with a n p using the recurrent relations we analyze the form of p for the both of random matrix families in the second part of the paper we consider the resolvents g a z of a n p and of n p and study the functions fn a z u 1 n k 1n exp ugkk a z in the limit n we derive closed equations that uniquely determine the limiting functio

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

self routing quantum sparse crossbar packet concentrators quantum switching networks are derived from conventional switching networks by replacing the classical switches by quantum switches we give the quantum circuit design and routing of an n m network called a quantum concentrator that can direct quantum bit packets in arbitrary quantum states from any of its k inputs to some of its k outputs where 1 k m n our designs are based on sparse crossbars which are rectangular grids of 2 2 crosspoints sparse crossbar concentrator structures with theoretically minimum crosspoint complexities for any values of n and m are well known but no self routing algorithms have been reported for such concentrators we transform two such families of optimal concentrators called fat slim and banded sparse crossbars into quantum networks and provide self routing algorithms for these families of concentrators in this process we extend the notion of packet concentration to a quantum network and design self routing quantum crosspoints from quantum gates we address issues critical to quantum operation like reversibility and localized self routing and give a rigorous proof that quantum fat slim and banded sparse crossbar concentrators are self routable the self routing algorithms described in the paper can be used for both quantum and classical sparse crossbar concentrators by the linearity property of all quantum systems

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