Asymptotic expansion of beta matrix models in the one-cut regime

TL;DR

This paper establishes a comprehensive 1/N asymptotic expansion for beta matrix models in the one-cut regime, using topological recursion and complex analysis, extending previous results to all orders.

Contribution

It introduces a method combining bounds and Schwinger-Dyson equations to derive all-order expansions for beta matrix models, generalizing prior work beyond Hermitian cases.

Findings

Proves existence of 1/N expansion to all orders in beta matrix models.

Recursively computes coefficients via topological recursion.

Extends previous results to general beta models and all orders.

Abstract

We prove the existence of a 1/N expansion to all orders in beta matrix models with a confining, off-critical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the "topological recursion" of Chekhov and Eynard. Our method relies on the combination of a priori bounds on the correlators and the study of Schwinger-Dyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following Boutet de Monvel, Pastur and Shcherbina, or for strictly convex potentials by using concentration of measure. Doing so, we extend the strategy of Guionnet and Maurel-Segala, from the hermitian models (beta = 2) and perturbative potentials, to general beta models. The existence of the first correction in 1/N has been considered previously by Johansson and more recently by Kriecherbauer and Shcherbina. Here, by taking similar hypotheses, we extend the result to all orders in 1/N.