Bifurcation Transitions and Expansions

It is believed in matrix model theory that when the eigenvalue density on one interval is split to a new density on two disjoint intervals, a phase transition occurs. The complexity for the mathematical details of this physical phenomenon comes not only from the elliptic integral calculations, but also from the organization of the parameters in the model. Generally, the elliptic integrals do not have simple analytic formulations for discussing the transition. The string equations can be applied to find the critical point for the transition from the parameter bifurcation, and the bifurcation clearly separates the different phases for analyzing the free energy. Based on the expansion method for elliptic integrals, the third-order bifurcation transition for the Hermitian matrix model with a general quartic potential is discussed in this chapter by applying the nonlinear relations obtained from the string equations. The density on multiple disjoint intervals for higher degree potential and the corresponding free energy are discussed in association with the Seiberg-Witten differential. In the symmetric cases for the quartic potential, the third-order phase transitions are explained with explicit formulations of the free energy function.