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 characterised 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 differential equations in the case β=1, and for the Gaussian ensemble, the case β=2/3. We apply the differential equations to give a systematic derivation of recurrences satisfied by the spectral moments and by the coefficients of their 1/N expansions, along with first-order differential equations for the coefficients of the 1/N expansions of the corresponding resolvents. We also present the form of the differential equations when scaled at the hard or soft edges.