$β$-ensembles and higher genera Catalan numbers

TL;DR

This work develops a comprehensive topological (large-$N$) expansion for the time-dependent free energy of β-deformed Gaussian and Wishart– Laguerre ensembles, encoding genus–dependent Catalan polynomials $C_{g,\nu}(β)$ whose coefficients are integers. Using nonlinear Virasoro constraints and a β↔1/β symmetry, the authors derive a cut-and-join recursion that fixes these polynomials, and show that at β=1 they recover the ordinary higher genus Catalan numbers. The analysis reveals a rich structure in which the β-deformation produces palindromic β-polynomials, a two-variable Schur basis representation, and an integer invariant set $n_{\nu,λ}$ with potential links to hypermaps and b-Hurwitz numbers. A parallel construction for the Wishart–Laguerre ensemble introduces an auxiliary φ-parameter and mirrors the Virasoro-based recursion, confirming a broad, integrality-rich framework for β-deformed matrix models and suggesting new combinatorial interpretations. Overall, the paper establishes a robust method to extract exact, integer-valued genus expansions across β-deformations and opens avenues for further topological and combinatorial insights in random matrix theory.

Abstract

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

Theorems & Definitions (13)

• Lemma 2.1
• proof
• Remark 3.1
• Proposition 5.1
• proof
• Remark 5.1
• Lemma 5.1
• proof
• Definition 5.1
• Remark 5.2