Correlation Functions of Complex Matrix Models

TL;DR

This paper derives explicit formulas for correlation functions in complex matrix models with specific potentials, using determinants of kernels from orthogonal polynomials, and explores their large N behavior and special limits.

Contribution

It introduces a determinant-based expression for correlation functions in complex matrix models with harmonic+Gaussian potentials, linking them to orthogonal polynomial kernels.

Findings

Explicit determinant formulas for correlation functions.

Large N ('t Hooft) expansion derived.

BMN limit analyzed for Gaussian potential.

Abstract

For a restricted class of potentials (harmonic+Gaussian potentials), we express the resolvent integral for the correlation functions of simple traces of powers of complex matrices of size $N$, in term of a determinant; this determinant is function of four kernels constructed from the orthogonal polynomials corresponding to the potential and from their Cauchy transform. The correlation functions are a sum of expressions attached to a set of fully packed oriented loops configurations; for rotational invariant systems, explicit expressions can be written for each configuration and more specifically for the Gaussian potential, we obtain the large $N$ expansion ('t Hooft expansion) and the so-called BMN limit.