Near extreme eigenvalues and the first gap of large random matrices

We study the phenomenon of “crowding” near the largest eigenvalue $\lambda_{max}$ of random $N \times N$ matrices belonging to the Gaussian Unitary Ensemble (GUE) of random matrix theory. We focus on two distinct quantities: (i) the density of states (DOS) near $\lambda_{\max}$, $\rho_{\text{DOS}}(r,N)$, which is the average density of eigenvalues located at a distance $r$ from $\lambda_{\max}$ and (ii) the probability density function (PDF) of the gap between the first two largest eigenvalues, $p_{\text{GAP}}(r,N)$. In the edge scaling limit where $r = \mathcal O(N^{-1/6})$, which is described by a double scaling limit of a system of unconventional orthogonal polynomials, we show that $\rho_{\text{DOS}}(r,N)$ and $p_{\text{GAP}}(r,N)$ are characterized by scaling functions which can be expressed in terms of the solution of a Lax pair associated to the Painlevé XXXIV equation. This provides an alternative and simpler expression for the gap distribution, which was recently studied by Witte, Bornemann and Forrester. Our expressions allow to obtain precise asymptotic behaviors of these scaling functions both for small and large arguments. I will also discuss extension of these results to the hard edge of Laguerre–Wishart random matrices.