CAREER: Random Matrices and Many-Body Systems

研究者的项目致力于随机矩阵理论的研究。这是一个非常活跃的领域，因为其他领域的许多问题都可以用数学方法表示为矩阵问题。例如：物理学中重原子的光谱，统计学中协方差矩阵的统计估计，量子通信中量子信道的容量，半导体的导电特性（安德森模型）。本项目旨在研究一些随机矩阵系综的基本性质，这将有助于揭示高度相关多体系统的本质。特别是，为了理解半导体和其他无序系统的导电特性，正如量子物理学中的安德森模型所解释的那样，本研究计划的一个主要项目旨在提供严格的证据，证明随机矩阵模型中出现了类似的（导电）特性。本课程旨在培养对概率论有兴趣的研究生和本科生，但存在一些问题。研究者还将举办研讨会，旨在促进在随机矩阵理论相关领域工作的研究生和本科生的向上专业发展，并促进这些学科之间的互动。研究者将重点研究许多不同类型的随机矩阵集合的特征值密度、局部特征值统计和特征向量分布的识别。本课题考虑的系综包括各向异性矩阵、带矩阵、i.i.d随机矩阵和一般非厄米矩阵。一个主要项目将集中在随机带矩阵的定位-离域猜想，这与无序量子系统的金属-绝缘体相变非常相似（即量子物理学中所谓的安德森猜想）。与常规Wigner矩阵不同，随机带矩阵不是平均场型系综。因此，这种系综更像是量子多体系统的现实模型，它涉及具有更多几何结构的量子态。另一个主要项目是研究非厄米矩阵（例如，i.i.d随机矩阵）。由于不能应用正则谱方法，对这类系综的研究远远弱于对随机厄米矩阵的研究。研究者将在这类矩阵上应用一些新思想。在本论文中，还设计了一些研究生和本科生的训练项目，包括证明更一般的随机矩阵集合的普适性现象。

The investigator's project is dedicated to a study of random matrix theory. It is a very active field, since many questions in other fields can be represented mathematically as matrix problems. For example: the spectra of heavy atoms in physics, statistical estimation of covariance matrices in statistics, capacity of quantum channels in quantum communication, conducting properties of semiconductors (Anderson models). This project aims to study the basic properties of some random matrix ensembles, which will shed some light on the nature of the highly correlated many-body system. In particular, for understanding the conducting properties of semiconductors and other disordered systems, as explained by the Anderson model in quantum physics, one main project in this research program is designed to provide rigorous proof that similar (conducting) property appears in the random matrix models. There are some problems in this program designed for training graduate and undergraduate students who have an interest in probability theory. The investigator will also run workshops designed to foster the upward professional development of graduate and undergraduate students working in fields related to random matrix theory as well as to stimulate interaction across barriers between these disciplines. The investigator will focus on the identification of the eigenvalue densities, the local eigenvalue statistics and the eigenvector distributions of many different classes of random matrix ensembles. The ensembles considered in this project include anisotropic matrices, band matrices, i.i.d. random matrices, and general non-Hermitian matrices. One main project will be focused on the localization-delocalization conjecture of random band matrices, which is very similar to the metal-insulator phase transition for disordered quantum systems (i.e., the so called Anderson conjecture in quantum physics). Different from regular Wigner matrices, random band matrices are not a mean-field type ensemble. Hence this ensemble is more like the realistic models of quantum many-body systems, which involve quantum states with more geometric structure. Another main project is a study of non-Hermitian matrices (e.g., i.i.d. random matrices). The study on this type of ensembles is much weaker than the study on random Hermitian matrices, since the regular spectrum method can not be applied. The investigator will apply some new ideas on these type of matrices. In this proposal, some projects are also designed for training graduate and undergraduate students, which include proving the universality phenomenon for more general random matrix ensembles.