An exact formula for general spectral correlation function of random Hermitian matrices

TL;DR

This paper derives an exact determinant formula for the general spectral correlation function of random Hermitian matrices, including products and ratios of characteristic polynomials, with broad applicability across ensembles.

Contribution

It introduces a novel exact formula involving both orthogonal polynomials and their Cauchy transforms, extending previous asymptotic results to a general setting.

Findings

Provides a determinant expression for correlation functions

Includes non-polynomial Cauchy transforms alongside orthogonal polynomials

Generalizes asymptotic formulas for GUE and chiral ensembles

Abstract

We have found an exact formula expressing a general correlation function containing both products and ratios of characteristic polynomials of random Hermitian matrices. The answer is given in the form of a determinant. An essential difference from the previously studied correlation functions (of products only) is the appearance of non-polynomial functions along with the orthogonal polynomials. These non-polynomial functions are the Cauchy transforms of the orthogonal polynomials. The result is valid for any ensemble of beta=2 symmetry class and generalizes recent asymptotic formulae obtained for GUE and its chiral counterpart by different methods..