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 ${\lambda }_{\max }$ 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 ${\lambda }_{\max }$, ${\rho }_{\mathrm{DOS}}(r,N)$, 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, $p_{\mathrm{GAP}}(r,N)$. 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.

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

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