Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies

We present a method to study phase transitions in the large N limit of matrix models using matched solutions of Whitham hierarchies. The endpoints of the eigenvalue spectrum as functions of the temperature are characterized both as solutions of hodograph equations and as solutions of a system of ordinary differential equations. In particular we show that the free energy of the matrix model is the quasiclassical tau-function of the associated hierarchy, and that critical processes in which the number of cuts changes in one unit are third-order phase transitions described by C1 matched solutions of Whitham hierarchies. The method is illustrated with the Bleher-Eynard model for the merging of two cuts. We show that this model involves also a birth of a cut.