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

Abstract

We characterize the phenomenon of "crowding" near the largest eigenvalue $λ_{\max}$ of random $N \times N$ matrices belonging to the Gaussian $β$-ensemble of random matrix theory, including in particular the Gaussian orthogonal ($β=1$), unitary ($β=2$) and symplectic ($β= 4$) ensembles. We focus on two distinct quantities: (i) the density of states (DOS) near $λ_{\max}$, $ρ_{\rm DOS}(r,N)$, which is the average density of eigenvalues located at a distance $r$ from $λ_{\max}$ (or the density of eigenvalues seen from $λ_{\max}$) 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 $β= 2$). We also discuss some applications of these two quantities to statistical physics models.

16 pages, 5 figures, contribution to the proceedings of the Workshop "Random Matrix Theory: Foundations and Applications" in Cracow, July 1-6 2014