Ratios of characteristic polynomials in complex matrix models

TL;DR

This paper derives explicit formulas for correlation functions involving ratios of characteristic polynomials in complex matrix models, extending previous results to complex eigenvalues using orthogonal polynomials.

Contribution

It provides new compact expressions for these correlation functions in terms of complex orthogonal polynomials and their Cauchy transforms, generalizing prior real eigenvalue results.

Findings

Explicit formulas for inverse powers and ratios of characteristic polynomials

Expressions in terms of orthogonal polynomials in the complex plane

Extension of previous real eigenvalue results to complex eigenvalues

Abstract

We compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues. Compact expressions are given in terms of orthogonal polynomials in the complex plane as well as their Cauchy transforms, generalizing previous expressions for real eigenvalues. We restrict ourselves to ratios of characteristic polynomials over their complex conjugate.