The Schur Expansion of Characteristic Polynomials and Random Matrices

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 $N$.