Near-Extreme Eigenvalues and the First Gap of Hermitian Random Matrices

We study the phenomenon of “crowding” near the largest eigenvalue λ _max of random N × N matrices belonging to the Gaussian Unitary Ensemble of random matrix theory. We focus on two distinct quantities: (i) the density of states (DOS) near λ _max , ρ _DOS(r,N) , which is the average density of eigenvalues located at a distance r from λ _max and (ii) the probability density function of the gap between the first two largest eigenvalues, p_GAP(r,N) . In the edge scaling limit where r = 𝒪(N^-1/6) , which is described by a double scaling limit of a system of unconventional orthogonal polynomials, we show that ρ _DOS(r,N) and p_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 et al. in Nonlinearity 26:1799, 2013 . Our expressions allow to obtain precise asymptotic behaviors of these scaling functions both for small and large arguments.