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

TL;DR

This paper proves that the eigenvalue density of beta-Laguerre ensembles converges to the Marchenko-Pastur law on very short scales, using simplified methods from previous work on local laws for beta-ensembles.

Contribution

It introduces a simplified proof of eigenvalue density convergence for beta-Laguerre ensembles on short scales, extending local law techniques.

Findings

Normalized trace of resolvent approximates MP distribution

Convergence of eigenvalue density to MP law on short scales

Simplified proof method for local eigenvalue statistics

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+\epsilon}. 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.