Moments of the eigenvalue densities and of the secular coefficients of $β$-ensembles

TL;DR

This work derives explicit finite-$N$ expressions for the moments of eigenvalue densities and the secular coefficients in Gaussian, Laguerre, and Jacobi $\beta$-ensembles by expanding power-sum observables in the Jack polynomial basis. Central to the method are the coefficients $\kappa_{(k)}^{\lambda}(\alpha)$ that express powers of traces in terms of Jack polynomials, together with ensemble-specific averages of $C_{\lambda}^{(\alpha)}$, computable via Selberg-type integrals. The authors provide closed forms for these averages in the Laguerre, Jacobi, and Gaussian cases, and also develop a framework for negative moments and higher-order correlations, illustrating the power and limitations of Jack-character techniques. In addition, they obtain explicit formulas for the secular coefficients in all three ensembles and discuss the open problem of fully determining joint moments in terms of Jack characters. Overall, the paper builds a concrete, finite-$N$ bridge between random-matrix moments and the algebra of symmetric functions, with potential applications to quantum chaos and related fields.

Abstract

We compute explicit formulae for the moments of the densities of the eigenvalues of the classical $β$-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.