Universalities in random matrix theory and quantum chaos

随机矩阵理论中的普适性表现在相关函数中，根据其对称性可分为若干普适性类。在这个研究项目中，我们研究了随机矩阵理论（RMT）与外部源。我们通过β参数来研究RMT， β21、β2和β 4分别对应于GOE、GUE和GSE。我们将Harish Chandra-Itzykson-Zuber的积分公式推广到β的一般值。研究了其与Jack多项式的关系。从边缘点处的相关函数得到黎曼曲面上的交点数，并在量子混沌和随机激光领域得到了广泛的应用。

The Universalities in random matrix theory appears in the correlation functions, and there are several universal classes depending upon the symmetries. In this research project, we have studied the random matrix theory (RMT) with an external source. We study RMT by a parameter β, which corresponds to GOE, GUE and GSE for β21,2 and 4, respectively. We have extended the integral formula of Harish Chandra-Itzykson-Zuber to a general value of β. The relation to Jack polynomial has been also studied. From the correlation function at the edge point, we obtain the intersection numbers on Riemann surfaces and applications to quantum chaos and random laser have been extensively studied.