CAREER: Random Matrices and Many-Body Systems

职业：随机矩阵和多体系统

• 批准号: 1552192
• 负责人: Jun Yin
• 金额: $ 40万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2018-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1552192&HistoricalAwards=false
• 关键词: CAREER; Random; Matrices; Many; Body

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.

研究者的项目致力于随机矩阵理论的研究。它是一个非常活跃的领域，因为其他领域的许多问题都可以在数学上表示为矩阵问题。举例来说：物理学中的重原子光谱，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，半导体的导电性质（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，这将有助于揭示高关联多体系统的本质。特别是，为了理解半导体和其他无序系统的导电特性，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，证明类似的（导电）特性出现在随机矩阵模型中。本专业是为培养对概率论感兴趣的研究生和本科生而设计的，但存在一些问题。调查员还将举办研讨会，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，并刺激这些学科之间的互动。 研究者将集中在许多不同类别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。在这个项目中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Hermitian矩阵。一个主要项目将集中在随机能带矩阵的局域化-离域化猜想上，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）。与规则Wigner矩阵不同，随机带矩阵不是平均场型系综。因此，这个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埃尔米特矩阵（例如，i.i.d.随机矩阵）。由于正则谱方法不能应用，对这类系综的研究远比随机厄米特矩阵的研究薄弱。研究人员将在这些类型的矩阵上应用一些新的想法。在这个方案中，还设计了一些培训研究生和本科生的项目，其中包括证明更一般的随机矩阵系综的普适性现象。