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

Abstract

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.

Large time zero temperature dynamics of the spherical p = 2-spin glass model of finite size

“…the eigenvalues of the coupling matrix J that are close to λ max [8,17]. More precisely, we will see that the observables mentioned above (energy density, response and correlation functions) can be written in terms of the density of eigenvalues "seen" from λ max , the so called density of states (DOS) ρ DOS (r, N ) defined as [18,19] …”

“…The DOS (9) was recently studied in detail for matrices belonging to the Gaussian Unitary Ensemble (GUE, corresponding to the Dyson index β = 2) [19], using semiclassical orthogonal polynomials, as well as for more general Gaussian β-ensembles [20], using mainly scaling arguments. In particular, it was shown that the behavior of the average DOS ρ DOS (r, N ) exhibits two distinct behaviors depending on whether r ∼ O(1), or r ∼ O(N −2/3 ).…”

“…In the former case ρ DOS (r, N ) is simply given by a shifted Wigner semicircle law (see also [17]), whereas in the latter ρ DOS (r, N ) takes a non-trivial scaling form reflecting the fluctuations at the edge of the Wigner semi-circle. This can be summarized as follows [19,20] …”

“…The scaling functionρ edge (x) is presently known exactly only for complex Hermitian random matrices belonging to GUE [19]. For general Gaussian β-ensembles, including the case β = 1 mostly relevant for the spin-glass model studied here (1), the exact evaluation ofρ edge (x) remains an open problem.…”

Spectral order statistics of Gaussian random matrices: Large deviations for trapped fermions and associated phase transitions

“…Details on the derivations, numerical checks and the outlook for future research will be provided elsewhere [36]. In the future, it will be interesting to study the order statistics for other ensembles and the crowding effects close to a specific eigenvalue in the bulk (see [50] for the first eigenvalue of GUE), as well as to investigate finite N corrections. In the Fermi gas picture (β = 2), our results provide the full statistics of particle number on a semi-infinite line (extending recent results [11,17,18]) and single-particle fluctuations in the bulk of a system of 1D fermions in a harmonic trap, including their LD tails.…”