The Goulden-Harer-Jackson matrix model

TL;DR

This paper derives a new formula for the Goulden-Harer-Jackson matrix model's partition function, unifies it with Penner models, and explores its implications for moduli space characteristics and critical points.

Contribution

It introduces an alternative partition function formula that encompasses Penner models and analyzes the model's free energy and critical behavior in relation to moduli space topology.

Findings

Unified formula for partition functions including Penner models

Expression for Euler characteristic involving Bernoulli polynomials

Same critical points for Goulden-Harer-Jackson and Penner models in continuum limit

Abstract

An alternative formula for the partition function of the Goulden-Harer-Jackson matrix model is derived, in which the Penner and the orthogonal Penner partition functions are special cases of this formula. Then the free energy that computes the parametrized Euler characteristic $\xi^s_g(\gamma)$ of the moduli spaces as yet an unidentified, for $g$ is odd, shows that the expression for $\xi^s_g(\gamma)$ contains the orbifold Euler characteristic of the moduli space of Riemann surfaces of genus $g$, with $s$ punctures for all parameters $\gamma$. The other contributions are written as a linear combinations of Bernoulli polynomials at rational arguments . It is also shown that in the continuum limit, both the Goulden-Harer-Jackson matrix model and the Penner model have the same critical points.