Universal behavior for averages of characteristic polynomials at the origin of the spectrum

TL;DR

This paper demonstrates that the kernels related to averages of characteristic polynomials for a class of unitary ensembles exhibit universal Bessel-function behavior at the spectrum's origin as matrix size grows, using Riemann-Hilbert analysis.

Contribution

It establishes the universal asymptotic behavior of these kernels at the spectrum's origin for a broad class of unitary ensembles, extending previous results.

Findings

Kernels exhibit universal Bessel behavior at the origin as n→∞

Method uses Riemann-Hilbert problem characterization of orthogonal polynomials

Asymptotic analysis applies steepest descent method to matrix Riemann-Hilbert problems

Abstract

It has been shown by Strahov and Fyodorov that averages of products and ratios of characteristic polynomials corresponding to Hermitian matrices of a unitary ensemble, involve kernels related to orthogonal polynomials and their Cauchy transforms. We will show that, for the unitary ensemble $\frac{1}{\hat Z_n}|\det M|^{2\alpha}e^{-nV(M)}dM$ of $n\times n$ Hermitian matrices, these kernels have universal behavior at the origin of the spectrum, as $n\to\infty$, in terms of Bessel functions. Our approach is based on the characterization of orthogonal polynomials together with their Cauchy transforms via a matrix Riemann-Hilbert problem, due to Fokas, Its and Kitaev, and on an application of the Deift/Zhou steepest descent method for matrix Riemann-Hilbert problems to obtain the asymptotic behavior of the Riemann-Hilbert problem.