Generalized Penner models to all genera

TL;DR

This paper provides a comprehensive analysis of the genus expansion and critical behavior of the generalized Penner model, including its relation to fermionic matrix models and multi-critical points across all genera.

Contribution

It introduces a complete description of the genus expansion for the generalized Penner model and explores its critical behavior and connections to fermionic matrix models through analytical continuation.

Findings

Critical points can be indexed by an integer m.

Multi-critical fermionic models share the same string susceptibility exponent as hermitian models.

Explicit construction of multi-critical points with alternating sign expansions.

Abstract

We give a complete description of the genus expansion of the one-cut solution to the generalized Penner model. The solution is presented in a form which allows us in a very straightforward manner to localize critical points and to investigate the scaling behaviour of the model in the vicinity of these points. We carry out an analysis of the critical behaviour to all genera addressing all types of multi-critical points. In certain regions of the coupling constant space the model must be defined via analytical continuation. We show in detail how this works for the Penner model. Using analytical continuation it is possible to reach the fermionic 1-matrix model. We show that the critical points of the fermionic 1-matrix model can be indexed by an integer, $m$, as it was the case for the ordinary hermitian 1-matrix model. Furthermore the $m$'th multi-critical fermionic model has to all genera the same value of $\gamma_{str}$ as the $m$'th multi-critical hermitian model. However, the coefficients of the topological expansion need not be the same in the two cases. We show explicitly how it is possible with a fermionic matrix model to reach a $m=2$ multi-critical point for which the topological expansion has alternating signs, but otherwise coincides with the usual Painlev\'{e} expansion.