Generalized Penner models to all genera

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 behavior of the model in the vicinity of these points. We carry out an analysis of the critical behavior to all genera addressing all types of multicritical 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 one-matrix model. We show that the critical points of the fermionic one-matrix model can be indexed by an integer m, as was the case for the ordinary Hermitian one-matrix model. Furthermore the mth multicritical fermionic model has to all genera the same value of ${\ensuremath{\gamma}}_{\mathrm{str}}$ as the mth multicritical 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 multicritical point for which the topological expansion has alternating signs, but otherwise coincides with the usual Painlev\'e expansion.