Convergence of the eigenvalue density for Laguerre beta ensembles on short scales

Philippe Sosoe,Percy Wong

doi:10.1214/ejp.v19-2638

Abstract

In this note, we prove that the normalized trace of the resolvent of the beta-Laguerre ensemble eigenvalues is close to the Stieltjes transform of the Marchenko-Pastur (MP) distribution with very high probability, for values of the imaginary part greater than $m^{1+\varepsilon}$. As an immediate corollary, we obtain convergence of the one-point density to the MP law on short scales. The proof serves to illustrate some simplifications of the method introduced in our previous work to prove a local semi-circle law for Gaussian beta-ensembles.

Highlights

Consider an m × n matrix X, whose entries are i.i.d. complex Gaussian random variables with mean 0 and variance E|Xij|2 = 1
The m × m matrix H = XX∗, the star ∗ denoting the conjugate transpose, is a Wishart matrix [23]. It is a classical result of random matrix theory that the distribution of the eigenvalues λ1, . . . , λm of H is given by the density f2(λ) on Rm: m f2(λ) = Z2−,m1 |λi − λj |2 · λni −me−β i
The argument in the present work is substantially simpler than the corresponding one in [20], where we proved a local convergence result on the scale m−1/2+ using a resolvent expansion and asymptotics for Hermite polynomials derived by the Riemann-Hilbert method

Summary

Introduction

Expansion, together with an iterative argument based on the Schur complement identity could be used to derive a local version of the semi-circle law for the Gaussian βensembles Tridiagonal models for these eigenvalue distributions appeared in [7]. The argument in the present work is substantially simpler than the corresponding one in [20], where we proved a local convergence result on the scale m−1/2+ using a resolvent expansion and asymptotics for Hermite polynomials derived by the Riemann-Hilbert method. The local result on the intermediate scale m−1/2+ in [20] was used as an input for an inductive argument to reach the scale m−1+ 1 It can be entirely replaced by the iteration, we have chosen to present the argument for Proposition 2.1 because it provides an elementary alternative to the Schur complement approach to proving the Marchenko-Pastur law. In contrast to the result in [20] at the time of publication, Theorem 1.1 appears to be new for general β

Tridiagonal models and resolvent expansion

Inductive argument

Some extensions