Averages of Characteristic Polynomials in Random Matrix Theory

TL;DR

This paper derives explicit pfaffian and determinantal formulas for averages of characteristic polynomials in classical random matrix ensembles, and analyzes their asymptotic behavior in the bulk scaling limit.

Contribution

It introduces a discrete approximation approach to compute these averages without relying on orthogonal polynomials, simplifying the derivation of classical correlation results.

Findings

Explicit pfaffian/determinantal formulas for averages obtained.

Bulk scaling limits derived from these formulas.

Classical correlation functions recovered from the new approach.

Abstract

We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with Gaussian weights. Classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulas by a simple computation. We employ a discrete approximation method: the problem is solved for discrete analogues of random matrix ensembles originating from representation theory, and then a limit transition is performed. Exact pfaffian/determinantal formulas for the discrete averages are proved using standard tools of linear algebra; no application of orthogonal or skew-orthogonal polynomials is needed.