Correlations of the characteristic polynomials in the Gaussian Unitary Ensemble or a singular Hankel determinant

TL;DR

This paper derives large n asymptotics for products of characteristic polynomial magnitudes in the Gaussian Unitary Ensemble, linking them to Hankel determinants with Fisher-Hartwig singularities, advancing understanding of eigenvalue correlations.

Contribution

It provides new asymptotic formulas for characteristic polynomial products and their relation to Hankel determinants with singularities, extending prior results in random matrix theory.

Findings

Asymptotics for characteristic polynomial products in GUE

Connection between these asymptotics and Hankel determinants with Fisher-Hartwig singularities

Enhanced understanding of eigenvalue correlations in random matrices

Abstract

We obtain large n asymptotics for products of powers of the absolute values of the characteristic polynomials in the Gaussian Unitary Ensemble of n\times n matrices. Our results can also be interpreted as asymptotics of the determinant of a Hankel matrix whose symbol is supported on the real line and possesses power-like (Fisher-Hartwig) singularities.