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arXiv · 2310.05626

$\beta$-ensembles and higher genera Catalan numbers

Luca Cassia·Vera Posch·Maxim Zabzine
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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.

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Luca Cassia, Vera Posch, Maxim Zabzine. 2023-10-09. $\beta$-ensembles and higher genera Catalan numbers. https://doi.org/10.1007/s11005-023-01764-x

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