The Schur Expansion of Characteristic Polynomials and Random Matrices

TL;DR

This paper introduces a new Schur polynomial-based method to compute exact correlators of characteristic polynomials in random matrix theory, applicable to general potentials and external sources, with novel results for inverse determinants and large M regimes.

Contribution

The paper develops a unified framework using Schur polynomial expansions to compute correlators in random matrices, including new finite N and large M results.

Findings

Exact finite N correlator for inverse determinants with external source

Method applicable to general potentials and external sources

Access to the large M > N regime for inverse determinant insertions

Abstract

We develop a new framework to compute the exact correlators of characteristic polynomials, and their inverses, in random matrix theory. Our results hold for general potentials and incorporate the effects of an external source. In matrix model realizations of string theory, these correspond to correlation functions of exponentiated "(anti-)branes" in a given background of "momentum branes". Our method relies on expanding the (inverse) determinants in terms of Schur polynomials, then re-summing their expectation values over the allowed representations of the symmetric group. Beyond unifying previous, seemingly disparate calculations, this powerful technique immediately delivers two new results: 1) the full finite $N$ answer for the correlator of inverse determinant insertions in the presence of a matrix source, and 2) access to an interesting, novel regime $M>N$, where the number of inverse determinant insertions $M$ exceeds the size of the matrix $N$.