Moments of the eigenvalue densities and of the secular coefficients of $\beta$-ensembles

TL;DR

This paper derives explicit formulas for the moments of eigenvalue densities and characteristic polynomial coefficients in classical β-ensembles, linking them to Jack polynomials and characters for finite matrix sizes.

Contribution

It provides new explicit finite-dimensional formulas for moments and characteristic coefficients in β-ensembles, connecting them with Jack polynomial theory.

Findings

Explicit formulas for eigenvalue density moments

Explicit formulas for characteristic polynomial coefficients

Connection to Jack polynomial averages

Abstract

We compute explicit formulae for the moments of the densities of the eigenvalues of the classical $\beta$-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters.