On high moments of strongly diluted large Wigner random matrices

We consider a dilute version of the Wigner ensemble of nxn random matrices $H$ and study the asymptotic behavior of their moments $M_{2s}$ in the limit of infinite $n$, $s$ and $\rho$, where $\rho$ is the dilution parameter. We show that in the asymptotic regime of the strong dilution, the moments $M_{2s}$ with $s=\chi\rho$ depend on the second and the fourth moments of the random entries $H_{ij}$ and do not depend on other even moments of $H_{ij}$. This fact can be regarded as an evidence of a new type of the universal behavior of the local eigenvalue distribution of strongly dilute random matrices at the border of the limiting spectrum. As a by-product of the proof, we describe a new kind of Catalan-type numbers related with the tree-type walks.Comment: 43 pages (version four: misprints corrected, discussion added, other minor modifications

Random matrices: Universality of local eigenvalue statistics up to the edge

This is a continuation of our earlier paper on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in that paper from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.Comment: 24 pages, no figures, to appear, Comm. Math. Phys. One new reference adde

A central limit theorem for the zeroes of the zeta function

On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Soshnikov and others in random matrix theory. In an appendix we put forward some general theorems regarding our knowledge of the zeta zeroes in the mesoscopic regime.Comment: 22 pages. Incorporates referees suggestions. Contains minor corrections to published versio

Quantum Diffusion and Delocalization for Band Matrices with General Distribution

We consider Hermitian and symmetric random band matrices $H$ in $d \geq 1$ dimensions. The matrix elements $H_{xy}$, indexed by $x,y \in \Lambda \subset \Z^d$, are independent and their variances satisfy \sigma_{xy}^2:=\E \abs{H_{xy}}^2 = W^{-d} f((x - y)/W) for some probability density $f$. We assume that the law of each matrix element $H_{xy}$ is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian $H$ is diffusive on time scales $t\ll W^{d/3}$. We also show that the localization length of the eigenvectors of $H$ is larger than a factor $W^{d/6}$ times the band width $W$. All results are uniform in the size \abs{\Lambda} of the matrix. This extends our recent result \cite{erdosknowles} to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying $\sum_x\sigma_{xy}^2=1$ for all $y$, the largest eigenvalue of $H$ is bounded with high probability by $2 + M^{-2/3 + \epsilon}$ for any $\epsilon > 0$, where M \deq 1 / (\max_{x,y} \sigma_{xy}^2).Comment: Corrected typos and some inaccuracies in appendix

On universality of local edge regime for the deformed Gaussian Unitary Ensemble

We consider the deformed Gaussian ensemble $H_n=H_n^{(0)}+M_n$ in which $H_n^{(0)}$ is a hermitian matrix (possibly random) and $M_n$ is the Gaussian unitary random matrix (GUE) independent of $H_n^{(0)}$. Assuming that the Normalized Counting Measure of $H_n^{(0)}$ converges weakly (in probability if random) to a non-random measure $N^{(0)}$ with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of $H_n$.Comment: 25 pages, 2 figure

Renormalized energy concentration in random matrices

We define a "renormalized energy" as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of [SS1,SS3]. Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number of charges. The functional is expected to be a good measure of disorder of a configuration of points. We give certain formulas for its expectation for general stationary random point processes. For the random matrix $\beta$-sine processes on the real line (beta=1,2,4), and Ginibre point process and zeros of Gaussian analytic functions process in the plane, we compute the expectation explicitly. Moreover, we prove that for these processes the variance of the renormalized energy vanishes, which shows concentration near the expected value. We also prove that the beta=2 sine process minimizes the renormalized energy in the class of determinantal point processes with translation invariant correlation kernels.Comment: last version, to appear in Communications in Mathematical Physic

On Eigenvalues of the sum of two random projections

We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N are two N -by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P_N + Q_N is not universal in the usual sense.Comment: 14 page

Mock-Gaussian Behaviour for Linear Statistics of Classical Compact Groups

We consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups. We compute their moments and find that the first few moments are Gaussian, whereas the limiting distribution is not. The precise number of Gaussian moments depends upon the particular statistic considered

Pattern densities in fluid dimer models

In this paper, we introduce a family of observables for the dimer model on a bi-periodic bipartite planar graph, called pattern density fields. We study the scaling limit of these objects for liquid and gaseous Gibbs measures of the dimer model, and prove that they converge to a linear combination of a derivative of the Gaussian massless free field and an independent white noise.Comment: 38 pages, 3 figure

Some Universal Properties for Restricted Trace Gaussian Orthogonal, Unitary and Symplectic Ensembles

Consider fixed and bounded trace Gaussian orthogonal, unitary and symplectic ensembles, closely related to Gaussian ensembles without any constraint. For three restricted trace Gaussian ensembles, we prove universal limits of correlation functions at zero and at the edge of the spectrum edge. In addition, by using the universal result in the bulk for fixed trace Gaussian unitary ensemble, which has been obtained by G$\ddot{o}$tze and Gordin, we also prove universal limits of correlation functions for bounded trace Gaussian unitary ensemble.Comment: 19pages,bounded trace Gaussian ensembles are adde