Universality for orthogonal and symplectic Laguerre-type ensembles

TL;DR

This paper proves the Universality Conjecture for orthogonal and symplectic Laguerre-type ensembles, establishing universal eigenvalue statistics at the bulk and edges of the spectrum.

Contribution

It extends universality results to beta=1 and beta=4 Laguerre ensembles, using orthogonal polynomial methods and asymptotic analysis.

Findings

Universal local eigenvalue statistics established

Correlation and gap probabilities characterized

Distribution of extreme eigenvalues determined

Abstract

We give a proof of the Universality Conjecture for orthogonal (beta=1) and symplectic (beta=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels K_{n,beta}, correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (beta=2) Laguerre-type ensembles have been proved by the fourth author in [23]. The varying weight case at the hard spectral edge was analyzed in [13] for beta=2: In this paper we do not consider varying weights. Our proof follows closely the work of the first two authors who showed in [7], [8] analogous results for Hermite-type ensembles. As in [7], [8] we use the version of the orthogonal polynomial method presented in [25], [22] to analyze the local eigenvalue statistics. The necessary asymptotic information on the Laguerre-type orthogonal polynomials is taken from [23].