Constructive Matrix Theory for Higher Order Interaction

TL;DR

This paper extends the constructive loop vertex expansion to higher order stable matrix models, introducing a new representation and forest expansion to prove the analyticity of the free energy series uniformly in matrix size.

Contribution

It introduces a novel representation and forest expansion method for higher order matrix models, establishing uniform analyticity of the free energy series.

Findings

Proves the perturbation series is analytic uniformly in matrix size N.

Extends the constructive loop vertex expansion to models with arbitrarily high order interactions.

Applicable to complex matrices; Hermitian case postponed.

Abstract

This paper provides an extension of the constructive loop vertex expansion to stable matrix models with interactions of arbitrarily high order. We introduce a new representation for such models, then perform a forest expansion on this representation. It allows to prove that the perturbation series of the free energy for such models is analytic in a domain uniform in the size N of the matrix. Our method applies to complex (rectangular) matrices. The extension to Hermitian square matrices, which was claimed wrongly in the first arXiv version of this paper, is postponed to a future study.