Correlations of the characteristic polynomials in the Gaussian unitary ensemble or a singular Hankel determinant

Abstract

We obtain large $n$ asymptotics for products of powers of the absolute values of the characteristic polynomials in the Gaussian unitary ensemble (GUE) 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