Phase transition of random system and matrix model

对于随机矩阵模型，研究了各种通用性及其交叉与随机系统相变的关系。研究了带外源的n点覆盖函数的随机矩阵、形状因子和水平间距分布，得到了它们的精确积分表示。在大N极限下，讨论了复盖函数的普适性。外部源为确定性哈密顿量与随机哈密顿量的交叉提供了基本途径。作为应用，本文对复杂随机矩阵和超导体涡旋激励进行了研究。研究了涡旋激振中从清涡到脏涡的交叉问题。与此相关，本文还研究了量子化霍尔效应的两种状态（自旋态），分别用于态的非化和电导率的研究。

For the random matrix model, various universality and their crossover have been studied in the relation to the phase transition of the random system. The random matrix with external source has been studied for the n-point covelation function, the form factor and the level spacing distribution, and the exact integral replesentation for them are obtained. In the large N limit, the universality of the covelation function has been discussed. The esternal source makes the basic approach for the crossover form the deterministic Hamiltonion to random Hamiltonian. As application, the complex random matrix and the superconductor vortex excitation have been unsidued. The crossover form the clear ease to the dirty case for the vortex excitation has been studied. Related to the cluiral in variouce, two states of the quanfized Hall effect(spin state) is also studied for the deusify of state and for the conductivity.