We use the tridiagonal matrix representation to derive a local semicircle law for Gaussian beta ensembles at the optimal level of $n^{-1+δ}$ for any $δ> 0$. Using a resolvent expansion, we first derive a semicircle law at the intermediate level of $n^{-1/2+δ}$; then an induction argument allows us to reach the optimal level. This result was obtained in a different setting, using different methods, by Bourgade, Erdös, and Yau and in Bao and Su. Our approach is new and could be extended to other tridiagonal models.
We derive and compare various forms of local semicircle laws for random matrices with exchangeable entries which exhibit correlations that decay at a very slow rate. In fact, any $l$-point correlation will decay at a rate of $N^{-l/2}$. We call our ensembles \emph{of Curie-Weiss type}, and Curie-Weiss($β$)-distributed entries are admissible as long as $β\leq 1$.
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.
Hochstättler, Kirsch, and Warzel showed that the semicircle law holds for generalized Curie-Weiss matrix ensembles at or above the critical temperature. We extend their result to the case of subcritical temperatures for which the correlations between the matrix entries are stronger. Nevertheless, one may use the concept of approximately uncorrelated ensembles that was first introduced in the paper mentioned above. In order to do so one needs to remove the average magnetization of the entries by an appropriate modification of the ensemble that turns out to be of rank 1 thus not changing the lim…
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.
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…
For two lacunary sequences $(M_{n,1})_{n\geq 2},(M_{n,2})_{n\geq 0}$ and suitable functions $f$ we introduce random matrix ensembles with \begin{equation*} X_{n,n'}=f(M_{n+n',1}x_1,M_{|n-n'|,2}x_2). \end{equation*} We prove weak convergence of the mean empirical eigenvalue distribution towards the semicircle law under some further number theoretic properties of the sequence $(M_{n,1})_{n\geq 1}$. Furthermore we give examples to show that even in this particular class of random matrix ensembles the asymptotic behaviour of the spectrum becomes delicate. We prove that the empirical spectral distr…
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 investigate the spectral distribution of random matrix ensembles with correlated entries. We consider symmetric matrices with real valued entries and stochastically independent diagonals. Along the diagonals the entries may be correlated. We show that under sufficiently nice moment conditions the empirical eigenvalue distribution converges almost surely weakly to the semi-circle law.
The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $β$ is allowed to vary with the matrix size $n$. In particular, we show that as $n \to \infty$ with $nβ\to \infty$, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.
We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [14] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \E \abs{h_{ij}}^2$. As a consequence, we prove the universality of the local $n$-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band …
We investigate the characteristic polynomials $φ_N$ of the Gaussian $β$-ensemble for general $β>0$ through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for $φ_N$ in terms of a Gaussian log--correlated field in order to ultimately deduce some of its fine asymptotic properties. We distinguish between different types of transfer matrices and analyze completely the hyperbolic regime of the recurrence. As a result, we obtain a new coupling between $φ_N(z)$ and a Gaussian analytic function with an error which is uniform for $z \in \mathbb{C}$ separated …
In this paper we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension $n$. In these cases the limit measure is given by the Marchenko-Pastur law and the semicircle law, respectively. For the weighted spectral measure we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.
We consider the Gaussian beta-ensemble when $β$ scales with $n$ the number of particles such that $\displaystyle{{n}^{-1}\ll β\ll 1}$. Under a certain regime for $β$, we show that the largest particle satisfies a large deviations principle in $\mathbb{R}$ with speed $nβ$ and explicit rate function. As a consequence, the largest particle converges in probability to $2$, the rightmost point of the semicircle law.
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 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 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…
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 …
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 $\varepsilon >0$.
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 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 consider random n\times n matrices of the form (XX*+YY*)^{-1/2}YY*(XX*+YY*)^{-1/2}, where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to \log n factors). This result is the analogue of the local …
We give a short proof of the free analogue of the Talagrand inequality for the transportation cost to the semicircular which was originally proved by Biane and Voiculescu. The proof is based on a convexity argument and is in the spirit of the original Talagrand's proof. We also discuss the convergence, fluctuations and large deviations of the energy of the eigenvalues of beta ensembles, which gives also yet another proof of the convergence of the eigenvalue distribution to the semicircle law.
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 compute exact asymptotic of the statistical density of random matrices belonging to the Generalized Gaussian orthogonal, unitary and symplectic ensembles such that there no eigenvalues in the interval $[σ, +\infty[$. In particular, we show that the probability that all the eigenvalues of an $(n\times n)$ random matrix are positive (negative) decreases for large $n$ as $\sim exp[-βθ(α)n^2]$ where the Dyson index $β$ characterizes the ensemble, $α$ is some extra parameter and the exponent $θ(α)$ is a function of $α$ which will be given explicitly. For $α=0$, $θ(0)= (\log 3)/4 = 0.274653...$ i…
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 the beta-Laguerre ensemble, a family of distributions generalizing the joint eigenvalue distribution of the Wishart random matrices. We show that the bulk scaling limit of these ensembles exists for all beta>0 for a general family of parameters and it is the same as the bulk scaling limit of the corresponding beta-Hermite ensemble.
We study the limiting behavior of Gaussian beta ensembles in the regime where $βn = const$ as $n \to \infty$. The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk statistics. (2) is an alternative proof of the result by F.~Benaych-Georges and S.~Péché (2015) with the explicit form of the intensity measure.
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…
Consider the ensembles of real symmetric Toeplitz matrices and real symmetric Hankel matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work on real symmetric Toeplitz matrices shows that the spectral measures, or densities of normalized eigenvalues, converge almost surely to a universal near-Gaussian distribution, while previous work on real symmetric Hankel matrices shows that the spectral measures converge almost surely to a universal non-unimodal distribution. Real symmetric Toeplitz m…
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…
We study the Gaussian Unitary Ensemble (GUE) using noncommutative geometry and the homological framework of the Batalin-Vilkovisky (BV) formalism. Coefficients of the correlation functions in the GUE with respect to the rank $N$ are described in terms of ribbon graph Feynman diagrams that then lead to a counting problem for the corresponding surfaces. The canonical relations provided by this homological setup determine a recurrence relation for these correlation functions. Using this recurrence relation and properties of the Catalan numbers, we determine the leading order behavior of the corre…
Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of $L$-functions. Many statistics in one can be interpreted in terms of quantities of the other; for example, zeros of $L$-functions correspond to eigenvalues of matrices, and values of $L$-functions to values of the characteristic polynomials. This correspondence has allowed RMT to successfully predict many number theory behaviors; however, there are some operations which to date have no RMT analogue. The motivation of this paper is to try and find an RMT equivalent to Rankin-Selbe…
We consider generalized Wigner ensembles and general beta-ensembles with analytic potentials for any beta larger than 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian beta-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal.
The distributions of $ N $-particle systems of Gaussian unitary ensembles converge to Sine$_2$ point processes under bulk-scaling limits. These scalings are parameterized by a macro-position $ θ$ in the support of the semicircle distribution. The limits are always Sine$_{2}$ point processes and independent of the macro-position $ θ$ up to the dilations of determinantal kernels. We prove a dynamical counter part of this fact. We prove that the solution of the $ N $-particle systems given by stochastic differential equations (SDEs) converges to the solution of the infinite-dimensional Dyson mode…
It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this paper, we prove analogs of Wigner's theorem for random matrices taken from all infinitesimal versions of classical symmetric spaces. This is a class of models which contains those studied by Wigner and Dyson, along with seven others arising in condensed matter physics. Like Wigner's, our results are universal in that they only depend on certain assumptions a…
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…
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 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 …
We prove the bulk universality of the $β$-ensembles with non-convex regular analytic potentials for any $β>0$. This removes the convexity assumption appeared in our earlier work. 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.
We show that the distribution of bulk spacings between pairs of adjacent eigenvalue real parts of a random matrix drawn from the complex elliptic Ginibre ensemble is asymptotically given by a generalization of the Gaudin-Mehta distribution, in the limit of weak non-Hermiticity. The same generalization is expressed in terms of an integro-differential Painlevé function and it is shown that the generalized Gaudin-Mehta distribution describes the crossover, with increasing degree of non-Hermiticity, from Gaudin-Mehta nearest-neighbor bulk statistics in the Gaussian Unitary Ensemble to Poisson gap …
We study the limiting behavior of the $k$-th eigenvalue $x_k$ of unitary invariant ensembles with Freud-type and uniform convex potentials. As both $k$ and $n-k$ tend to infinity, we obtain Gaussian fluctuations for $x_k$ in the bulk and soft edge cases, respectively. Multi-dimensional central limit theorems, as well as moderate deviations, are also proved. This work generalizes earlier results in the GUE and unitary invariant ensembles with monomial potentials of even degree. In particular, we obtain the precise asymptotics of corresponding Christoffel-Darboux kernels as well.
The bead process introduced by Boutillier is a countable interlacing of the determinantal sine-kernel point processes. We construct the bead process for general sine beta processes as an infinite dimensional Markov chain whose transition mechanism is explicitly described. We show that this process is the microscopic scaling limit in the bulk of the Hermite beta corner process introduced by Gorin and Shkolnikov, generalizing the process of the minors of the Gaussian unitary and orthogonal ensembles. In order to prove our results, we use bounds on the variance of the point counting of the circul…
The Airy line ensemble is a central object in random matrix theory and last passage percolation defined by a determinantal formula. The goal of this paper is to provide a set of tools which allow for precise probabilistic analysis of the Airy line ensemble. The two main theorems are a representation in terms of independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines. Along the way, we give tail bounds and moduli of continuity for nonintersecting Brownian ensembles, and a quick proof of tightness for Dyson's Brownian motion converging to t…
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…
Consider the classical Gaussian unitary ensemble of size $N$ and the real Wishart ensemble $W_N(n,I)$. In the limits as $N \to \infty$ and $N/n \to γ> 0$, the expected number of eigenvalues that exit the upper bulk edge is less than one, 0.031 and 0.170 respectively, the latter number being independent of $γ$. These statements are consequences of quantitative bounds on tail sums of eigenvalues outside the bulk which are established here for applications in high dimensional covariance matrix estimation.
We consider the adjacency matrix of the ensemble of Erdős-Rényi random graphs which consists of graphs on $N$ vertices in which each edge occurs independently with probability $p$. We prove that in the regime $pN \gg 1$ 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.
Universal limits for the eigenvalue correlation functions in the bulk of the spectrum are shown for a class of nondeterminantal random matrices known as the fixed trace ensemble.
This paper is the second chapter of three of the author's undergraduate thesis. In this paper, we consider the random matrix ensemble given by $(d_b, d_w)$-regular graphs on $M$ black vertices and $N$ white vertices, where $d_b \in [N^γ, N^{2/3 - γ}]$ for any $γ> 0$. We simultaneously prove that the bulk eigenvalue correlation statistics for both normalized adjacency matrices and their corresponding covariance matrices are stable for short times. Combined with an ergodicity analysis of the Dyson Brownian motion in another paper, this proves universality of bulk eigenvalue correlation statistic…
In the setting of generic $β$-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the following consequences. (i) The optimal rigidity scale of the ordered particles is of order $(\log N)/N$ in the bulk of the spectrum. (ii) Fluctuations of the particles satisfy a central limit theorem with covariance corresponding to a logarithmically correlated field; in particular each particle in the bulk fluctuates on scale $\sqrt{\log N}/N$. (iii) The logarithm of the electric potential also s…
