Linear differential equations for the resolvents of the classical matrix ensembles

Abstract

The spectral density for random matrix β ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of β, which for even β is a polynomial of degree β(N - 1). In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover, the spectral density itself, can be characterized as the solution of a linear differential equation of degree β + 1. This equation, and its companion for the resolvent, are given explicitly for β = 2 and 4 for all three classical cases, and also for β = 6 in the Gaussian case. Known dualities for the spectral moments relating β to 4/β then imply corresponding ..