On Universality of Bulk Local Regime of the Deformed Laguerre Ensemble

TL;DR

This paper proves the universality of local eigenvalue statistics in the bulk spectrum of the deformed Laguerre Ensemble, under certain conditions on the deformation matrix, extending understanding of spectral behavior in random matrix theory.

Contribution

It establishes the universality of bulk local eigenvalue statistics for the deformed Laguerre Ensemble with general deformation matrices, under weak convergence assumptions.

Findings

Universality holds for the bulk eigenvalue statistics.

Results apply to possibly random deformation matrices.

Convergence is established under weak assumptions on the spectral measure.

Abstract

We consider the deformed Laguerre Ensemble $H_n=\dfrac{1}{m}\Sigma_n^{1/2}A_{m,n}A_{m,n}^*\Sigma_n^{1/2}$ in which $\Sigma_n$ is a positive hermitian matrix (possibly random) and $A_{m,n}$ is a $n\times m$ complex Gaussian random matrix (independent of $\Sigma_n$), $\dfrac{m}{n}\to c>1$. Assuming that the Normalized Counting Measure of $\Sigma_n$ converges weakly (in probability) to a non-random measure $N^{(0)}$ with a bounded support we prove the universality of the local eigenvalue statistics in the bulk of the limiting spectrum of $H_n$.