Characteristic Polynomials of Complex Random Matrix Models

TL;DR

This paper derives a general formula for the expectation of products of characteristic polynomials in complex random matrix models, extending known results for hermitian matrices and exploring universality in large-N limits.

Contribution

It provides a determinant-based expression for characteristic polynomial expectations in complex matrices with general weights, generalizing the Christoffel formula to the complex plane.

Findings

Derived explicit formulas for finite-N complex matrix models.

Established universality of characteristic polynomials in the large-N weak limit.

Discussed implications for the BMN large-N limit.

Abstract

We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite-N, where N is the size of the matrix. We give some explicit examples at finite-N for specific weight functions. The characteristic polynomials in the large-N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large-N limit.