Linear differential equations for the resolvents of the classical matrix ensembles
Anas A Rahman, Peter J Forrester
RANDOM MATRICES-THEORY AND APPLICATIONS | WORLD SCIENTIFIC PUBL CO PTE LTD | Published : 2021
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 ..View full abstract
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Awarded by Australian Research Council
The authors are grateful to the referees for their valuable feedback. The work of PJF was partially supported by the Australian Research Council Grant DP170102028 and the ARC Centre of Excellence for Mathematical and Statistical Frontiers, and that of AAR by the Australian Government Research Training Program Scholarship and the ARC Centre of Excellence for Mathematical and Statistical Frontiers.