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The partition function of the Penner matrix model for both positive andnegative values of the coupling constant can be explicitly written in terms ofthe Barnes G function. In this paper we show that for negative values of thecoupling constant this partition function can also be represented as theproduct of an holomorphic matrix integral by a nontrivial oscillatory functionof n. We show that the planar limit of the free energy with 't Hooft sequencesdoes not exist. Therefore we use a certain modification that usesKuijlaars-McLaughlin sequences instead of 't Hooft sequences and leads to awell-defined planar free energy and to an associated two-dimensional phasespace. We describe the different configurations of complex saddle points of theholomorphic matrix integral both to the left and to the right of the criticalpoint, and interpret the phase transitions in terms of processes of gapclosing, eigenvalue tunneling, and Bose condensation.

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