The density of eigenvalues seen from the soft edge of random matrices in the Gaussian beta-ensembles

TL;DR

This paper investigates the local eigenvalue density and gap distribution near the largest eigenvalue in Gaussian beta-ensembles, revealing detailed crowding phenomena and extending previous Hermitian case results.

Contribution

It generalizes the analysis of eigenvalue crowding near the maximum to all beta-ensembles, including non-Hermitian cases, with heuristic and numerical methods.

Findings

Density of states near the largest eigenvalue characterized

Distribution of the gap between top two eigenvalues analyzed

Results applicable to statistical physics models

Abstract

We characterize the phenomenon of "crowding" near the largest eigenvalue $\lambda_{\max}$ of random $N \times N$ matrices belonging to the Gaussian $\beta$-ensemble of random matrix theory, including in particular the Gaussian orthogonal ($\beta=1$), unitary ($\beta=2$) and symplectic ($\beta = 4$) ensembles. We focus on two distinct quantities: (i) the density of states (DOS) near $\lambda_{\max}$, $\rho_{\rm DOS}(r,N)$, which is the average density of eigenvalues located at a distance $r$ from $\lambda_{\max}$ (or the density of eigenvalues seen from $\lambda_{\max}$) and (ii) the probability density function of the gap between the first two largest eigenvalues, $p_{\rm GAP}(r,N)$. Using heuristic arguments as well as well numerical simulations, we generalize our recent exact analytical study of the Hermitian case (corresponding to $\beta = 2$). We also discuss some applications of these two quantities to statistical physics models.