We study the phenomenon of "crowding" near the largest eigenvalue λ max of random N × 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 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, Bornemann and Forrester in Nonlinearity 26, 1799 (2013). Our expressions allow to obtain precise asymptotic behaviors of these scaling functions both for small and large arguments.

Abstract. We revisit the long time dynamics of the spherical fully connected p = 2-spin glass model when the number of spins N is large but finite. At T = 0 where the system is in a (trivial) spin-glass phase, and on long time scale t O(N 2/3 ) we show that the behavior of physical observables, like the energy, correlation and response functions, is controlled by the density of near-extreme eigenvalues at the edge of the spectrum of the coupling matrix J, and are thus non self-averaging. We show that the late time decay of these observables, once averaged over the disorder, is controlled by new universal exponents which we compute exactly.arXiv:1507.08520v2 [cond-mat.dis-nn]