Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

TL;DR

This paper establishes universal formulas for correlation functions of characteristic polynomials of Hermitian random matrices using Riemann-Hilbert techniques, revealing universal behavior across invariant ensembles.

Contribution

It introduces a Riemann-Hilbert approach to derive exact determinant formulas for correlation functions involving characteristic polynomials, including their asymptotics and moments.

Findings

Correlation functions are governed by three types of integrable kernels.

Exact determinant formulas for correlation functions are obtained.

Universal behavior of moments of characteristic polynomials is demonstrated.

Abstract

We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of $\beta=2$ symmetry class.