Higher Genus Correlators for the Complex Matrix Model

TL;DR

This work develops an iterative framework to compute all multi-loop correlators for the complex matrix model at arbitrary genus using only the first loop equation, applicable to a general potential and depending on a finite set of moments. It casts the genus-$g$ contribution as a finite linear combination of basis functions, with coefficients constrained by precise homogeneity relations, and proves the conjecture by induction, obtaining explicit genus-2 and genus-3 results. The authors also establish an equivalence between the complex matrix model and an external-field problem with a logarithmic potential, connecting to Kontsevich–Penner-type structures. In the double-scaling limit, the results align with the Hermitian matrix model up to a simple scaling factor, opening a path to rigorous comparisons and higher-genus analysis beyond planar approximations.

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.