Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles

Abstract

The spectral density for random matrix $\beta$ ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of $\beta$, which for even $\beta$ is a polynomial of degree $\beta (N-1)$. 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 $\beta +1$. This equation, and its companion for the resolvent, are given explicitly for $\beta =2$ and $4$ for all three classical cases, and also for $\beta =6$ in the Gaussian case. Known dualities for the spectral moments relating $\beta$ to $4/\beta$ then imply corresponding differential equations in the case $\beta =1$, and for the Gaussian ensemble, the case $\beta =2/3$. 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.

arXiv comment43 pages

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Cited by in corpus (3)

• Differential identities for the structure function of some random matrix ensembles
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Relations between moments for the Jacobi and Cauchy random matrix ensembles