Moments of characteristic polynomials for classical $\beta$ ensembles

TL;DR

This paper derives simplified asymptotic formulas for moments of characteristic polynomials in classical beta ensembles, establishing universality and extending results to cases with proportional exponents.

Contribution

It provides simplified derivations, error bounds, and extends asymptotic results to Laguerre and Jacobi ensembles with proportional exponents.

Findings

Asymptotic formulas for moments with error bounds

Universality of the constant term across ensembles

Extension to cases with proportional exponents in Laguerre and Jacobi ensembles

Abstract

For random matrix ensembles with unitary symmetry, there is interest in the large $N$ form of the moments of the absolute value of the characteristic polynomial for their relevance to the Riemann zeta function on the critical line, and to Fisher-Hartwig asymptotics in the theory of Toeplitz determinants. The constant (with respect to $N$) in this asymptotic expansion, involving the Barnes $G$ function, is most relevant to the first of these, while the algebraic term (in $N$) and the functional dependence on the power are of primary interest in the latter. Desrosiers and Liu [20] have obtained the analogous expansions for the classical Gaussian, Laguerre and Jacobi $\beta$ ensembles in the case of even moments. We give simplified working of these results -- which requires the use of duality formulas and the use of steepest descents for multidimensional integrals -- providing too an error bound on the resulting asymptotic expressions. The universality of the constant term with respect to an earlier result known for the circular $\beta$ ensemble is established, which requires writing it in a Barnes $G$ function form, while the functional dependence on the powers is related to that appearing in Gaussian fluctuation formulas for linear statistics. In the Laguerre and Jacobi cases our working can be extended to the circumstance when the exponents in the weight function are (strictly) proportional to $N$, giving results not previously available in the literature.