Universality for Orthogonal and Symplectic Laguerre-type Ensembles

We give a proof of the Universality Conjecture for orthogonal (β = 1) and symplectic (β = 4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. They concern the appropriately rescaled kernels K n,β , correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (β = 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 β = 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].