Higher Genus Correlators for the Complex Matrix Model

TL;DR

This paper introduces an iterative method to compute multi-loop correlators at any genus for the complex matrix model, applicable to general potentials and independent of specific couplings, with explicit results up to genus three.

Contribution

It presents a novel iterative scheme based on loop equations for calculating correlators in the complex matrix model at arbitrary genus and potential.

Findings

Explicit generating functional up to genus three.

Method works for any potential without explicit coupling dependence.

Model equivalent to an external field problem with a logarithmic potential.

Abstract

We describe an iterative scheme which allows us to calculate any multi-loop correlator for the complex matrix model to any genus using only the first in the chain of loop equations. The method works for a completely general potential and the results contain no explicit reference to the couplings. The genus $g$ contribution to the $m$--loop correlator depends on a finite number of parameters, namely at most $4g-2+m$. We find the generating functional explicitly up to genus three. We show as well that the model is equivalent to an external field problem for the complex matrix model with a logarithmic potential.