Schur function expansion in non-Hermitian ensembles and averages of characteristic polynomials

TL;DR

This paper develops a new method using Schur function expansions to analyze k-point correlators of characteristic polynomials in non-Hermitian random matrix ensembles, providing explicit formulas and probabilistic interpretations.

Contribution

It introduces a novel character expansion technique for non-Hermitian ensembles and demonstrates how to re-sum these expansions, linking correlators to Young diagram distributions.

Findings

Explicit formulas for correlators in terms of determinants and Pfaffians.

Extension of the method to truncations of random unitary matrices.

Probabilistic interpretation connecting correlators to Young diagram top rows.

Abstract

We study $k$-point correlators of characteristic polynomials in non-Hermitian ensembles of random matrices, focusing on the real, complex and quaternion $N \times N$ Ginibre ensembles. Our approach is based on the technique of character expansions, which expresses the correlator as a sum over partitions involving Schur functions. We show how to re-sum the expansions in terms of representations which interchange the roles of $N$ and $k$. We also provide a probabilistic interpretation of the character expansion analogous to the Schur measure, linking the correlators to the distribution of the top row in certain Young diagrams. In more specific examples we evaluate these expressions explicitly in terms of $k \times k$ determinants or Pfaffians. We show that our approach extends to other ensembles, such as truncations of random unitary matrices.