$\beta$-ensembles and higher genera Catalan numbers

TL;DR

This paper develops formulas for the large N expansion of connected correlators in $eta$-deformed matrix models, revealing new combinatorial structures related to higher genus Catalan numbers and their properties under parameter transformations.

Contribution

It introduces a recursive approach to compute expansion coefficients in $eta$-deformed models and defines higher genus Catalan polynomials with integer coefficients for all $eta$.

Findings

The formulas satisfy known $eta o 1/eta$ symmetry.

Recursion formulas for expansion coefficients are derived.

Higher genus Catalan polynomials have integer coefficients for all $eta$.

Abstract

We propose formulas for the large $N$ expansion of the generating function of connected correlators of the $\beta$-deformed Gaussian and Wishart-Laguerre matrix models. We show that our proposal satisfies the known transformation properties under the exchange of $\beta$ with $1/\beta$ and, using Virasoro constraints, we derive a recursion formula for the coefficients of the expansion. In the undeformed limit $\beta=1$, these coefficients are integers and they have the combinatorial interpretation of generalized Catalan numbers. For generic $\beta$, we define the higher genus Catalan polynomials $C_{g,\nu}(\beta)$ whose coefficients are integer numbers.