# Phase space and phase transitions in the Penner matrix model with negative coupling constant

## TL;DR

This paper investigates the phase structure of the Penner matrix model with negative coupling, revealing complex saddle point configurations and phase transitions related to eigenvalue phenomena, using modified sequences for a well-defined free energy.

## Contribution

It introduces a new approach using Kuijlaars-McLaughlin sequences to define a consistent planar free energy for negative coupling in the Penner model.

## Key findings

- Planar free energy does not exist with 't Hooft sequences for negative coupling.
- Modified sequences yield a well-defined planar free energy and phase space.
- Identifies phase transitions as gap closing, eigenvalue tunneling, and Bose condensation.

## Abstract

The partition function of the Penner matrix model for both positive and negative values of the coupling constant can be explicitly written in terms of the Barnes G function. In this paper we show that for negative values of the coupling constant this partition function can also be represented as the product of an holomorphic matrix integral by a nontrivial oscillatory function of n. We show that the planar limit of the free energy with 't Hooft sequences does not exist. Therefore we use a certain modification that uses Kuijlaars-McLaughlin sequences instead of 't Hooft sequences and leads to a well-defined planar free energy and to an associated two-dimensional phase space. We describe the different configurations of complex saddle points of the holomorphic matrix integral both to the left and to the right of the critical point, and interpret the phase transitions in terms of processes of gap closing, eigenvalue tunneling, and Bose condensation.