Moments of characteristic polynomials for classical $β$ ensembles

Bo-Jian Shen, Peter J. Forrester

Published 2025-02-11Version 1

In the case of the circular unitary ensemble of Haar distributed unitary matrices, the large $N$ form of the moments of the absolute value of the characteristic polynomial is known to be of relevance to the moments of the Riemann zeta function on the critical line, and further to relate 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 known to appear in the analogous expansion for the moments of the characteristic polynomial in other classical ensembles with unitary symmetry, for example the Gaussian unitary ensemble. Using certain duality formulas for averages over powers of characteristic polynomials, this same effect is studied for classical $\beta$ ensembles, which is tractable in the case of even moments. For a given value of $\beta$, a distinguished factor in the asymptotic expansion can be identified which is universal with respect to the particular $\beta$ ensemble.