Spectral Properties of Random Matrices

这个项目将推进我们对随机矩阵的了解，随机矩阵是一组作为许多高维系统范例的算子。虽然高斯分布普遍出现的概率模型表现出独立性，随机矩阵统计出现相关系统，使其应用范围最近有了很大的扩展。例如，可积系统，增长模型，解析数论和量子力学表现出随机矩阵统计。此外，随机矩阵技术提高了我们对数值分析和深度学习网络的理论理解。随机矩阵的谱分析最近提出了概率（随机游动和耦合）、分析（均匀化）和数学物理（回路方程）的方法，以证明特征值和特征向量的普适性。PI将进一步开发这样的工具，用于以下问题：（1）带矩阵的离域。PI旨在理解超越平均场模型的普遍性。一个关键的问题涉及一维带矩阵，直到“安德森”过渡。最近的一种方法涉及到平均场约化和与量子唯一遍历性的联系，它将进一步发展;（2）极值统计的普适性。 这包括极端的光谱间距，大值的特征多项式，波动的个别特征值和更普遍的方面的对数相关的随机场;（3）库仑气体，高斯自由场和高斯乘性混沌。最近的进展使二维库仑气体的电场收敛到高斯自由场。一个自然的推广涉及高斯乘性混沌的特征多项式的收敛性;（4）非厄米特随机矩阵的特征向量。物理学家在20世纪90年代预测了Ginibre系综本征向量之间重叠的典型大小。这是最近升级到一个明确的限制分布的对角线重叠，并相关的非对角线重叠。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project will advance our knowledge of random matrices, a set of operators that serves as a paradigm for many high-dimensional systems. While the Gaussian distribution emerges universally from probabilistic models exhibiting independence, random matrix statistics appear from correlated systems, so that the scope of their applications have considerably expanded recently. For example, integrable systems, growth models, analytic number theory and quantum mechanics exhibit random matrix statistics. Moreover, random matrix techniques have improved our theoretical understanding of numerical analysis and deep learning networks. Methods from probability (random walks and coupling), analysis (homogenization) and mathematical physics (loop equations) were recently advanced in connection with spectral analysis of random matrices, to prove eigenvalues and eigenvectors universality. The PI will further develop such tools for the following problems: (1) Delocalization for band matrices. The PI aims at understanding universality beyond mean field models. One key question concerns band matrices in dimension 1 up to the ``Anderson'' transition. A recent approach to this problem involves a mean field reduction and the connection with quantum unique ergodicity, and it will be developed further; (2) Universality of extremal statistics. This includes extreme spectral spacings, large values of the characteristic polynomial, fluctuations of individual eigenvalues and more generally universal aspects of log-correlated random fields;(3) Coulomb gases, the Gaussian free field and Gaussian multiplicative chaos. Recent progress gave convergence of the electric field of 2D Coulomb gases to the Gaussian free field. A natural extension concerns convergence of the characteristic polynomial to Gaussian multiplicative chaos;(4) Eigenvectors of non-Hermitian random matrices. Physicists predicted in the 1990s the typical size of overlaps between eigenvectors from the Ginibre ensemble. This was recently upgraded to an explicit limiting distribution of diagonal overlaps, and to correlations of off-diagonal overlaps. Extension of this new integrability to the joint law of all overlaps will be studied.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.