多割矩阵模型中的相变与Whitham族的匹配解
Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies
摘要：
提出了一种利用Whitham族的匹配解研究矩阵模型在大N极限下的相变的方法。作为温度的函数的本征值谱的端点的特征在于作为速端方程的解决方案和作为一个系统的常微分方程的解决方案。特别是，我们表明，矩阵模型的自由能是准经典的相关层次的τ-函数，和关键的过程中，在一个单元中的切割数的变化是三阶相变所描述的C1匹配的解决方案的Whitham层次。该方法说明了与Bleher-Eynard模型合并的两个削减。我们表明，这个模型也涉及到一个削减的诞生。
英文摘要：
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.
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