Random Matrix Theory and its Applications

1. 研究了具有对数相互作用的一维粒子系统的统计力学。自由能对温度的依赖是普遍的。对于与经典正交多项式相关的系综，证明了水平分布涨落的普适性。对于圆形系综，相关函数用正交或斜正交多项式表示。对于与经典正交多项式相关的系综，证明了系综的遍历性。一维离散薛定谔方程被认为是水平统计的一个模型。水平统计不存在局部波动。实空间中对应于Jastrow波函数的分布函数用正交或斜正交多项式表示。表明了谱在边缘处的非普适性，并用参数描述谱，表明谱在边缘处具有弱普适性。讨论了扰动作用下哈密顿矩阵的运动。与经典正交多项式相关的随机矩阵系综，特别是与指数型分布相对应的随机矩阵系综，被证明是介观系统中电导率的合适模型。讨论了经典Calogero模型与矩阵模型的关系。

1. A statistical mechanics of one-dimensional particle system with logarithmic interactions is studied. Temperature dependence of the free energy is shown to be universal.2. For ensembles related to classical orthogonal polynomials, the universality of fluctuations of level distributions is proved.3. For the circular ensemble, correlation functions are expressed in terms of orthogonal or skew-orthogonal polynomials.4. Ergodicity of ensembles is proved for ensembles related to classical orthogonal polynomials.5. One-dimensional discrete Schrodinger equation is considered as a model for level statistics. There exists no local fluctuation of level statistics.6. Distribution functions in real space corresponding to Jastrow wavefunctions are expressed in terms of orthogonal or skew-orthogonal polynomials.7. Non-universality of the spectrum at the edge is shown and the spectrum is described by a parameter, which indicates a weak universality at the edge.8. Motion of hamiltonian matrix under a perturbation is discussed.9. Random matrix ensembles related to classical orthogonal polynomials, in particular ones corresponding to exponential-type distributions, are shown to be suitable models for conductivity in mesoscopic systems.10. Relations betweem classical Calogero model and matrix model are discussed.