Monotone Hurwitz numbers and the HCIZ integral

TL;DR

This paper establishes a link between the convergence of the HCIZ free energy and the non-vanishing of the HCIZ integral near zero, using combinatorial and complex analysis methods.

Contribution

It provides a new combinatorial and analytic framework to understand the convergence properties of the HCIZ free energy.

Findings

Complex convergence of HCIZ free energy is equivalent to non-vanishing of HCIZ integral near zero.

Develops a combinatorial model for the Maclaurin coefficients of the HCIZ integral.

Uses classical complex-analytic techniques to establish the main result.

Abstract

In this article, we prove that the complex convergence of the HCIZ free energy is equivalent to the non-vanishing of the HCIZ integral in a neighbourhood of $z=0$. Our approach is based on a combinatorial model for the Maclaurin coefficients of the HCIZ integral together with classical complex-analytic techniques.