New recursive residue formulas for the topological expansion of the Cauchy Matrix Model

TL;DR

This paper develops new recursive residue formulas to extend the topological expansion of the Cauchy matrix model to cases with non-simple, non-singular branch points where two branch points merge without pinching cycles.

Contribution

It introduces novel recursive formulas that handle complex branch point configurations in the topological expansion of the Cauchy matrix model, expanding previous limitations.

Findings

Derived formulas for non-simple branch points

Extended topological expansion to new branch point configurations

Provided tools for analyzing more complex matrix models

Abstract

In a recent work [1] we consider the topological expansion for the non-mixed observables (including the free energy) for the formal Cauchy matrix model. The only restriction in [1] was the fact that all the branch points have to be simple. This excludes a very interesting situation not encountered in the literature before, namely the case in which two branch points merge in such a way that no cycle is pinched. In this work we focus on this situation and derive new formulas that apply to non-simple and non-singular branch-points.