The Penner matrix model and c = 1 strings

This paper reports on the steepest decent solution of the Penner matrix model which has a one-cut eigenvalue support. Criticality results when the two branch points of this support coalesce. The support is then a closed contour in the complex eigenvalue plane. Simple generalizations of the Penner model have multi-cut solutions. For these models, the eigenvalue support at criticality is also a closed contour, but consisting of several cuts. The authors solve the simplest such model, which the authors call the KT model, in the double-scaling limit. Its free energy is a Legendre transform of the free energy of the c = 1 string compactified to the critical radius of the Kosterlitz-Thouless phase transition