Random Matrices and Disordered Systems

随机矩阵和无序系统

• 批准号: 1307444
• 负责人: Horng-Tzer Yau
• 金额: $ 36万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1307444&HistoricalAwards=false
• 关键词: Random; Matrices; Disordered; Systems

In the recent work, the universality conjecture for the local spectral statistics was solved for both Winger ensembles and beta ensembles. The goal of this proposal is to extend this understanding to large classes of random matrices. The objective is to understand the grand vision of Wigner, which asserts that local spectral statistics of large correlated quantum systems are universal. Specifically, the following projects are proposed: 1. The spectral properties of Wigner matrices with random potentials. These are toy models for random Schrodinger operators on lattice. 2. The statistics of both eigenvalues and eigenvectors in band and sparse matrices. In addition to these two projects, local statistics at the level of individual eigenvalues will also be studied. More specifically, the intention is to look into the following problems: 3. The single gap universality, i.e., identify the probability distribution of a single gap in random matrices. 4. The universality of local statistics at a fixed energy. 5. Identify the Gaussian fluctuations of a single eigenvalue in Wigner matrices, i.e., extending Gustavsson's result for GUE to all Wigner matrices. 6. Establish the edge universality via Dyson's Brownian motion. These problems will be approached using the Helffer and Sjostrand representation of correlation functions and the parabolic regularity in the work of Caffarelli, Chan and Vasseur. Eugene Wigner's grand vision asserts that local spectral statistics of large correlated quantum systems are universal. In our recent work, we solved the universality conjecture for the local spectral statistics for both Wigner ensembles and beta ensembles. The goal of this proposal is to extend these findings to large classes of random matrices and expand our understanding of Wigner's vision.

在最近的工作中，求解了Winger系综和beta系综的局部谱统计量的普适性猜想。本建议的目标是将这种理解扩展到大的随机矩阵类。目标是理解维格纳的宏大愿景，他断言大型相关量子系统的局部谱统计是普遍的。具体而言，提出以下项目：具有随机势的Wigner矩阵的谱性质。这些是晶格上随机薛定谔算子的玩具模型。2. 特征值和特征向量在频带和稀疏矩阵中的统计量。除了这两个项目外，还将研究个体特征值水平上的局部统计。更具体地说，目的是研究以下问题：单间隙通用性，即识别随机矩阵中单间隙的概率分布。4. 定能局部统计量的普适性。5. 识别Wigner矩阵中单个特征值的高斯涨落，即将Gustavsson关于GUE的结果推广到所有Wigner矩阵。6. 通过戴森布朗运动建立边的普适性。这些问题将使用Helffer和Sjostrand的相关函数表示和Caffarelli， Chan和Vasseur的工作中的抛物正则性来处理。尤金·维格纳（Eugene Wigner）的宏大愿景断言，大型相关量子系统的局部谱统计是普遍的。在我们最近的工作中，我们解决了Wigner系综和beta系综的局部谱统计量的普适性猜想。本提案的目标是将这些发现扩展到大型随机矩阵，并扩展我们对维格纳愿景的理解。