Random Matrices and Disordered Systems

在最近的工作中，解决了Winger系综和Beta系综的局部谱统计的普适性猜想。这个建议的目标是将这种理解扩展到大类随机矩阵。我们的目标是理解维格纳的宏伟愿景， 它断言 大关联量子系统的局域谱统计是普适的。具体而言，提出以下项目：1. Wigner矩阵的谱性质 随机电位。这些是随机薛定谔算子在格上的玩具模型。2.带矩阵和稀疏矩阵的特征值和特征向量的统计。 除了这两个项目外，还将研究个别特征值一级的地方统计。更具体地说，目的是研究以下问题：3。单间隙普遍性，即，确定随机矩阵中单个间隙的概率分布。4.在固定能量下局部统计量的普适性。5.识别维格纳矩阵中单个特征值的高斯波动，即，将Gustavsson关于GUE的结果推广到所有Wigner矩阵。6.通过Dyson的布朗运动建立边的普适性。这些问题将采用Helffer和Sjostrand表示的相关函数和抛物线规律的工作Caffarelli，陈和Vasseur。尤金·维格纳的宏伟愿景断言，大关联量子系统的局域谱统计是普适的。在最近的工作中，我们解决了Wigner系综和Beta系综的局部谱统计的普适性猜想。这个建议的目标是将这些发现扩展到大类随机矩阵，并扩大我们对维格纳的视野的理解。

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.