Analyticity results for the cumulants in a random matrix model

TL;DR

This paper develops convergent and analytic expansions for cumulants in random matrix models, demonstrating their Borel summability and analyticity within a specific complex domain, thus advancing the mathematical understanding of these models.

Contribution

It introduces alternative convergent expansions for cumulants and their remainders, establishing their analyticity and Borel summability in the complex plane.

Findings

Cumulants are analytic inside a cardioid domain in the complex plane.

Explicit convergent expansions for cumulants and their remainders are provided.

Cumulants are proven to be Borel summable at the origin.

Abstract

The generating function of the cumulants in random matrix models, as well as the cumulants themselves, can be expanded as asymptotic (divergent) series indexed by maps. While at fixed genus the sums over maps converge, the sums over genera do not. In this paper we obtain alternative expansions both for the generating function and for the cumulants that cure this problem. We provide explicit and convergent expansions for the cumulants, for the remainders of their perturbative expansion (in the size of the maps) and for the remainders of their topological expansion (in the genus of the maps). We show that any cumulant is an analytic function inside a cardioid domain in the complex plane and we prove that any cumulant is Borel summable at the origin.