Generalized Penner models and multicritical behavior.

Abstract

In this paper, we are interested in the critical behavior of generalized Penner models at $t\ensuremath{\sim}\ensuremath{-}1+\frac{\ensuremath{\mu}}{N}$ where the topological expansion for the free energy develops logarithmic singularities: $\ensuremath{\Gamma}\ensuremath{\sim}\ensuremath{-}({\ensuremath{\chi}}_{0}{\ensuremath{\mu}}^{2}\mathrm{ln}\ensuremath{\mu}+{\ensuremath{\chi}}_{1}\mathrm{ln}\ensuremath{\mu}+\ensuremath{\cdots})$. We demonstrate that these criticalities can best be characterized by the fact that the large-$N$ generating function becomes meromorphic with a single pole term of unit residue, $F(z)\ensuremath{\rightarrow}\frac{1}{(z\ensuremath{-}a)}$, where $a$ is the location of the sink. For a one-band eigenvalue distribution, we identify multicritical potentials; we find that none of these can be associated with the $c=1$ string compactified at an integral multiple of the self-dual radius. We also give an exact solution to the Gaussian Penner model and explicity demonstrate that, at criticality, this solution does not correspond to a $c=1$ string compactified at twice the self-dual radius.