These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero expectation and constant variance. We state and prove the local semicircle law, which says that the eigenvalue distribution of a Wigner matrix is close to Wigner's semicircle distribution, down to spectral scales containing slightly more than one eigenvalue. This local semicircle law is formulated using the Green function, whose individual entries are controlled by large deviation bounds.
We then discuss three applications of the local semicircle law: first, complete delocalization of the eigenvectors, stating that with high probability the eigenvectors are approximately flat; second, rigidity of the eigenvalues, giving large deviation bounds on the locations of the individual eigenvalues; third, a comparison argument for the local eigenvalue statistics in the bulk spectrum, showing that the local eigenvalue statistics of two Wigner matrices coincide provided the first four moments of their entries coincide. We also sketch further applications to eigenvalues near the spectral edge, and to the distribution of eigenvectors.
We consider Hermitian random matrices of the form $H = W + λV$, where $W$ is a Wigner matrix and $V$ a diagonal random matrix independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $λ\sim 1$ so that the eigenvalues of $W$ and $λV$ are of the same order in the bulk of the spectrum. In this paper, we prove for a large class of diagonal matrices $V$ that the local deformed semicircle law holds for $H$, which is an analogous result to the local semicircle law for Wigner matrices. We also prove complete delocalization of eigenvectors and other results about t…
We analyse the spectrum of additive finite-rank deformations of $N \times N$ Wigner matrices $H$. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue $d_i$ of the deformation crosses a critical value $\pm 1$. This transition happens on the scale $|d_i| - 1 \sim N^{-1/3}$. We allow the eigenvalues $d_i$ of the deformation to depend on $N$ under the condition $|\abs{d_i} - 1| \geq (\log N)^{C \log \log N} N^{-1/3}$. We make no assumptions on the eigenvectors of the deformation. In the limit $N \to \infty$, we …
We introduce kernel estimators for the semicircle law. In this first part of our general theory on the estimators, we prove the consistency and conduct simulation study to show the performance of the estimators. We also point out that Wigner's semicircle law for our new estimators and the classical empirical spectral distributions is still true when the elements of Wigner matrices don't have finite variances but are in the domain of attraction of normal law.
Recently we considered a class of random matrices obtained by choosing distinct codewords at random from linear codes over finite fields and proved that under some natural algebraic conditions their empirical spectral distribution converges to Wigner's semicircle law as the length of the codes goes to infinity. One of the conditions is that the dual distance of the codes is at least 5. In this paper, employing more advanced techniques related to Stieltjes transform, we show that the dual distance being at least 5 is sufficient to ensure the convergence, and the convergence rate is of the form …
We derive the joint asymptotic distribution of the outlier eigenvalues of an additively deformed Wigner matrix $H$. Our only assumptions on the deformation are that its rank be fixed and its norm bounded. Our results extend those of [The isotropic semicircle law and deformation of Wigner matrices. Preprint] by admitting overlapping outliers and by computing the joint distribution of all outliers. In particular, we give a complete description of the failure of universality first observed in [Ann. Probab. 37 (2009) 1-47; Ann. Inst. Henri Poincaré Probab. Stat. 48 (1013) 107-133; Free convolution…
We consider real symmetric and complex Hermitian random matrices with the additional symmetry $h_{xy}=h_{N-x,N-y}$. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble (GOE). It also occurs as the flip matrix model - an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law o…
We extend the random characteristics approach to Wigner matrices whose entries are not required to have a normal distribution. As an application, we give a simple and fully dynamical proof of the weak local semicircle law in the bulk.
Let $β>0$ and consider an $n$-point process $λ_1, λ_2,..., λ_n$ from Hermite $β$ ensemble on the real line $\mathbb{R}$. Dumitriu and Edelman discovered a tri-diagonal matrix model and established the global Wigner semicircle law for normalized empirical measures. In this paper we prove that the average number of states in a small interval in the bulk converges in probability when the length of the interval is larger than $\sqrt {\log n}$, i.e., local semicircle law holds. And the number of positive states in $(0,\infty)$ is proved to fluctuate normally around its mean $n/2$ with variance like…
We consider ensembles of $N \times N$ 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 {\it arbitrarily} small intervals converges to the semicircle law, as $N$ tends to infinity.
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.
In the present paper we give two alternate proofs of the well known theorem that the empirical distribution of the appropriately normalized roots of the $n^{th}$ monic Hermite polynomial $H_n$ converges weakly to the semicircle law, which is also the weak limit of the empirical distribution of appropriately normalized eigenvalues of a Wigner matrix. In the first proof -- based on the recursion satisfied by the Hermite polynomials -- we show that the generating function of the moments of roots of $H_n$ is convergent and it satisfies a fixed point equation, which is also satisfied by $c(z^2)$, w…
This in an introduction to random matrix theory, giving an impression of some of the most important aspects of this modern subject. In particular, it covers the basic combinatorial and analytic theory around Wigner's semicircle law, featuring also concentration phenomena, and the Tracy-Widom distribution of the largest eigenvalue. The circular law and a discussion of Voiculescu's multivariate extension of the semicircle law, as an appetizer for free probability theory, also make an appearance. The material here was presented in the winter term 2019/20 at Saarland University in 24 lectures of 9…
We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from the Wigner semicircle law.
We consider random Hermitian matrices with independent upper triangular entries. Wigner's semicircle law says that under certain additional assumptions, the empirical spectral distribution converges to the semicircle distribution. We characterize convergence to semicircle in terms of the variances of the entries, under natural assumptions such as the Lindeberg condition. The result extends to certain matrices with entries having infinite second moments. As a corollary, another characterization of semicircle convergence is given in terms of convergence in distribution of the row sums to the sta…
We extend the proof of the local semicircle law for generalized Wigner matrices given in [4] to the case when the matrix of variances has an eigenvalue $ -1 $. In particular, this result provides a short proof of the optimal local Marchenko-Pastur law at the hard edge (i.e. around zero) for sample covariance matrices $ \boldsymbol{\mathrm{X}}^\ast \boldsymbol{\mathrm{X}} $, where the variances of the entries of $ \boldsymbol{\mathrm{X}} $ may vary.
In this paper we study ensembles of random symmetric matrices $\X_n = {X_{ij}}_{i,j = 1}^n$ with dependent entries such that $\E X_{ij} = 0$, $\E X_{ij}^2 = σ_{ij}^2$, where $σ_{ij}$ may be different numbers. Assuming that the average of the normalized sums of variances in each row converges to one and Lindeberg condition holds we prove that the empirical spectral distribution of eigenvalues converges to Wigner's semicircle law.
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 in J. Stat. Phys., v.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.
We consider $N\times N$ Hermitian Wigner random matrices $H$ where the probability density for each matrix element is given by the density $ν(x)= e^{- U(x)}$. We prove that the eigenvalue statistics in the bulk is given by Dyson sine kernel provided that $U \in C^6(\RR)$ with at most polynomially growing derivatives and $ν(x) \le C e^{- C |x|}$ for $x$ large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.
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 [22,10], 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 bou…
Consider $N\times N$ hermitian or symmetric random matrices $H$ with independent entries, where the distribution of the $(i,j)$ matrix element is given by the probability measure $ν_{ij}$ with zero expectation and with variance $σ_{ij}^2$. We assume that the variances satisfy the normalization condition $\sum_{i} σ^2_{ij} = 1$ for all $j$ and that there is a positive constant $c$ such that $c\le N σ_{ij}^2 \le c^{-1}$. We further assume that the probability distributions $ν_{ij}$ have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution…
We study the local semicircle law for Gaussian $β$-ensembles at the edge of the spectrum. We prove that at the almost optimal level of $n^{-2/3+ε}$, the local semicircle law holds for all $β\geq 1$ at the edge. The proof of the main theorem relies on the calculation of the moments of the tridiagonal model of Gaussian $β$-ensembles up to the $p_n$-moment where $p_n = O(n^{2/3-ε})$. The result is the analogous to the result of Sinai and Soshnikov for Wigner matrices, but the combinatorics involved in the calculations are different.
Our main result is a local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices, modeled on the local semicircle law. Our approach is to adapt some techniques from one of the recent papers of Erdös-Yau-Yin. We also use an algebraic description of the law of the anticommutator of free semicircular variables due to Nica-Speicher, a self-adjointness-preserving variant of the linearization trick due to Haagerup-Schultz-Thorbjørnsen, and the Schwinger-Dyson equation. A byproduct of our work is a relatively simple deterministic version of the local s…
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 $η= \Im (z)$ 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…
Spectral measures of Wigner matrices are investigated. The Wigner semicircle law for spectral measures is proved. Regard this as the law of large number, the central limit theorem for moments spectral measure is also derived. The proof is based on moment method and combinatorial method.
We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\to\infty$ (with a speed at least logarithmic in $N$), the density of eigenvalues of the Erdős-Rényi 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 …
We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with upper triangular entries being i.i.d. random variables with mean zero and unit variance. We additionally suppose that $\mathbb E |X_{11}|^{4 + δ} =: μ_{4+δ} < \infty$ for some $δ> 0$. The aim of this paper is to significantly extend recent result of the authors [18] and show that with high probability the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-\frac{1}{2}} {\bf X}$ and Wigner's semicircle law is of order $(nv)^{-1} \log n$, where $v$ denotes the di…
We consider the joint distribution of eigenvalue clusters of the Wigner ensemble separated by macroscopic distances (i.e., on the same scale as the difference between the edges of the semicircle law). We prove that under an averaging condition, the correlation function governing any finite collection of clusters converges to that of independence point processes. The proof relies heavily on the machinery developed by Erdos, Ramirez, Schlein, and Yau.
We study the spectral properties of the dilute Wigner random real symmetric n-dimensional matrices H such that the entries H(i,j) take zero value with probability 1-p/n. We prove that under rather general conditions on the probability distribution of H(i,j) the semicircle law is valid for the dilute Wigner ensemble in the limit of infinite n and p. In the second part of the paper we study the leading term of the correlation function of the resolvent G(z) of H with large enough Im z in the limit of infinite n and p such that 3/5 log n <log p < log n. We show that this leading term, when conside…
We compute an asymptotic expansion with precision 1/n of the moments of the expected empirical spectral measure of Wigner matrices of size n with independent centered entries. We interpret this expansion as the moments of the addition of the semicircle law and 1/n times an explicit signed measured with null total mass. This signed measure depends only on the second and fourth moments of the entries.
Let $x_1, \dots, x_n$ be $n$ independent and identically distributed random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1, \dots, x_n$. We prove that for $\ell \in \mathbb{N}$ fixed as $n \rightarrow \infty$, the $(n-\ell)-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$ in a compact interval,$${n^{\ell/2}} \frac{\ell!}{n!} \cdot p_n^{(n-\ell)}\left( \frac{x}{\sqrt{n}}\right) \rightarrow He_{\ell}(x + γ_n),$$ where $He_{\ell}$ is the $\ell-$th probabilists' Hermite polynomial a…
In this paper, we study the complex Wigner matrices $M_n=\frac{1}{\sqrt{n}}W_n$ whose eigenvalues are typically in the interval $[-2,2]$. Let $λ_1\leq λ_2...\leqλ_n$ be the ordered eigenvalues of $M_n$. Under the assumption of four matching moments with the Gaussian Unitary Ensemble(GUE), for test function $f$ 4-times continuously differentiable on an open interval including $[-2,2]$, we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold $u$ in the bulk of the Wigner semicircle law as $\mathcal{A}_n[f; u]=\…
We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ in which the upper triangular entries are independent identically distributed random variables with mean zero and unit variance. We additionally suppose that $\mathbb E |X_{11}|^{4 + δ} =: μ_4 < \infty$ for some $δ> 0$. Under these conditions we show that the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-\frac{1}{2}} {\bf X}$ and Wigner's semicircle law is of order $(nv)^{-1}$, where $v$ is the distance in the complex plane to the real line. Furthermore we out…
We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with upper triangular entries being independent identically distributed random variables with mean zero and unit variance. We additionally suppose that $\mathbb E |X_{11}|^{4 + δ} =: μ_{4+δ} < C$ for some $δ> 0$ and some absolute constant $C$. Under these conditions we show that the typical Kolmogorov distance between the empirical spectral distribution function of eigenvalues of $n^{-1/2} {\bf X}$ and Wigner's semicircle law is of order $1/n$ up to some logarithmic correction factor. As a direct consequence of this result we…
We consider linear spectral statistics of the form $\mathrm{tr} ( φ(H))$ for test functions $φ$ of low regularity and Wigner matrices $H$ with smooth entry distribution. We show that for functions $φ$ in the Sobolev space $H^{1/2+\varepsilon}$ or the space $C^{1/2+\varepsilon}$, that are supported within the spectral bulk of the semicircle distribution, these linear spectral statistics have asymptotic Gaussian fluctuations with the same variance as in the CLT for functions of higher regularity, for any $\varepsilon >0$.
This note presents some central limit theorems for the eigenvalue counting function of Wigner matrices in the form of suitable translations of results by Gustavsson and O'Rourke on the limiting behavior of eigenvalues inside the bulk of the semicircle law for Gaussian matrices. The theorems are then extended to large families of Wigner matrices by the Tao and Vu Four Moment Theorem. Similar results are developed for covariance matrices.
The aim of this paper is to provide a novel proof for the Local Semicircle Law for the Wigner ensemble. The core of the proof is the intensive use of the algebraic structure that arises, i.e. resolvent expansions and resolvent identities. On the analytic side, concentration of measure results and high probability bounds are used. The conclusion is obtained using a bootstrapping argument that provides information about the change of the bounds from large to small scales. This approach leads to a new and shorter proof of the Weak Local Law for Wigner Matrices, that exploits heavily the algebraic…
This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erdös, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empir…
Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the distribution of the $(i,j)$ matrix element is given by a probability measure $ν_{ij}$ with a subexponential decay. Let $σ_{ij}^2$ be the variance for the probability measure $ν_{ij}$ with the normalization property that $\sum_{i} σ^2_{ij} = 1$ for all $j$. Under essentially the only condition that $c\le N σ_{ij}^2 \le c^{-1}$ for some constant $c>0$, we prove that, in the limit $N \to \infty$, the eigenvalue spacing statistics of $H$ in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal en…
The universality for the eigenvalue spacing statistics of generalized Wigner matrices was established in our previous work \cite{EYY} under certain conditions on the probability distributions of the matrix elements. A major class of probability measures excluded in \cite{EYY} are the Bernoulli measures. In this paper, we extend the universality result of \cite{EYY} to include the Bernoulli measures so that the only restrictions on the probability distributions of the matrix elements are the subexponential decay and the normalization condition that the variances in each row sum up to one. The n…
We consider asymptotics of orthogonal polynomial ensembles, in the macroscopic and mesoscopic scales. We prove both global and local laws of large numbers (analogous to the recently proven local semicircle law for Wigner matrices) under fairly weak conditions on the underlying measure $μ$. Our main tools are a general concentration inequality for determinantal point processes with a kernel that is a self-adjoint projection, and a strengthening of the Nevai condition from the theory of orthogonal polynomials.
We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the probability distribution for each matrix element is given by a measure $ν$ with a subexponential decay. Our main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic Ensemble (GSE), re…
Since E.P.Wigner (1958) established his famous semicircle law, lots of attention has been paid by physicists, probabilists and statisticians to study the asymptotic properties of the largest eigenvalues for random matrices. Bai and Yin (1988) obtained the necessary and sufficient conditions for the strong convergence of the extreme eigenvalues of a Wigner matrix. In this paper, we consider the case of quaternion self-dual Hermitian matrices. We prove the necessary and sufficient conditions for the strong convergence of extreme eigenvalues of quaternion self-dual Hermitian matrices correspondin…
We study the distribution (w.r.t. the vacuum state) of family of partial sums Sm of position operators on weakly monotone Fock space. We show that any single operator has the Wigner law, and an arbitrary family of them (with the index set linearly ordered) is a collection of monotone independent random variables. It turns out that our problem equivalently consists in finding the m-fold monotone convolution of the semicircle law. For m = 2 we compute the explicit distribution. For any m > 2 we give the moments of the measure, and show it is absolutely continuous and compactly supported on a sym…
Consider a random matrix of the form $W_n = M_n + D_n$, where $M_n$ is a Wigner matrix and $D_n$ is a real deterministic diagonal matrix ($D_n$ is commonly referred to as an external source in the mathematical physics literature). We study the universality of the local eigenvalue statistics of $W_n$ for a general class of Wigner matrices $M_n$ and diagonal matrices $D_n$. Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrate…
We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner-type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes transforms explicitly with weaker assumptions on the convergence of variance profiles than previous results. As applications, we give a new proof of the semicircle law for generalized Wigner matrices and determine the limiting spectral distributions for three sparse inhomogeneous random graph models with sparsity $ω(1/n)$: inhomogeneous random graphs with roughly equ…
We consider the real eigenvalues of an $(N \times N)$ real elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $τ_N\in [0,1]$. In the almost-Hermitian regime where $1-τ_N=Θ(N^{-1})$, we obtain the large-$N$ expansion of the mean and the variance of the number of the real eigenvalues. Furthermore, we derive the limiting empirical distributions of the real eigenvalues, which interpolate the Wigner semicircle law and the uniform distribution, the restriction of the elliptic law on the real axis. Our proofs are based on the skew-orthogonal polynomial representa…
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_{ij} = \mathbb{E}\, |h_{ij}|^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 [1]. 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 Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix models, the eigenvalue distributions are given by a log-gas with inverse temperature $β= 1, 2, 4$, corresponding to the orthogonal, unitary and symplectic ensembles. For $β\not \in \{1, 2, 4\}$, there is no matrix model behind this model, but the statistical physics interpretation of the log-gas is still valid for all $β> 0$. The universality conjecture for invariant ensembles …
We consider $n\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\mathbb {P}(|X_{ij}|\geq t)$ behave like $e^{-at^α}$ for some $a>0$ and $α\in(0,2)$. We establish a large deviation principle for the empirical spectral measure of $X/\sqrt{n}$ with speed $n^{1+α/2}$ with a good rate function $J(μ)$ that is finite only if $μ$ is of the form $μ=μ_{\mathrm{sc}}\boxplusν$ for some probability measure $ν$ on $\mathbb {R}$, where $\boxplus$ denotes the free convolution and $μ_{\mathrm{sc}}$ is Wigner's semicircle law. We obtain explicit expressions for $J(μ_{\mathrm{sc…
We review the application of the notion of local convergence on locally finite randomly rooted graphs, known as Benjamini-Schramm convergence, to the calculation of the global eigenvalue density of random matrices from the beta-Gaussian and beta-Laguerre ensembles. By regarding a random matrix as the weighted adjacency matrix of a graph, and choosing the root of such a graph with uniform probability, one can use the Benjamini-Schramm limit to produce the spectral measure of the adjacency operator of the limiting graph. We illustrate how the Wigner semicircle law and the Marchenko-Pastur law ar…
We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding $\ell_2$-operator norms. The key to our analysis is a generalisation of a classic result by d…
In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices $(M_N)_N$ defined by $M_N=W_N/\sqrt{N}+A_N$ where $W_N$ is an $N\times N$ Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincaré inequality. The matrix $A_N$ is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of $A_N$ are sufficiently far from zero, the corresponding eigenvalues of $M_N$ almost surely exit the limiting semic…
