Analytic Methods for the Random Matrix Universality Class

调查员将研究随机矩阵，一个领域起源于尤金维格纳的想法，随机矩阵模型的光谱行为的物理系统。多年来，随机矩阵的应用已经超越了纯数学，并被用于统计学，计算机科学，电信和更广泛的信息网络。这个研究项目的重点是普适性--极限谱统计仅取决于对称类型而不取决于底层系统的其他细节的现象。随机矩阵统计现在在可积系统、增长模型和数论的许多方面都有应用。在过去的几年里，已经取得了很大的进展，并且已经建立了越来越多的一类随机矩阵的普遍性。本计画主要探讨以下研究课题：（1）随机带矩阵的普适性与量子唯一遍历性。我们的动机是试图通过这些随机薛定谔算子的玩具模型来接近安德森转变。(2)通过最近引入的一种新的随机游动--特征向量矩流，在非微扰状态下对特征向量进行微扰分析。(3)对数相关场和随机谱。这包括维格纳矩阵和β系综的特征值的个体波动。(4)随机矩阵的极值统计，通过最大和最小间隙，以及特征多项式的极值。(5)非厄米随机矩阵理论的研究，二维库仑气体和高斯自由场之间的联系。为了理解维格纳的观点，在他们最近的固定能量普适性和本征向量普适性的证明中，研究者和合作者为上述项目开发了新的有趣工具。其中包括动态随机环境中的随机游动，耦合方法，以及含时随机系数偏微分方程的均匀化理论。

The investigator will study random matrices, a field originating with Eugene Wigner's idea that random matrices model spectral behavior of physical systems. Over the years random matrices have found applications beyond pure mathematics, and are being used in statistics, computer science, telecommunications, and more generally information networks. This research project's focus is universality -- the phenomenon that the limiting spectral statistics depend only on the symmetry type and not on other details of the underlying system. Random matrix statistics now find use in many aspects of integrable systems, growth models, and number theory. Deep progress has been achieved in the past years, and universality has been established for a growing class of random matrices. This project will advance understanding in this fundamental area.This project explores the following research topics:(1) Universality and quantum unique ergodicity for random band matrices. The motivation is to try to approach the Anderson transition via these toy models for random Schrodinger operators on a lattice.(2) Perturbative analysis of eigenvectors in a non-perturbative regime, via the eigenvector moment flow, a new random walk introduced recently. (3) Log-correlated fields and random spectra. This includes individual fluctuations of eigenvalues of Wigner matrices and beta ensembles. (4) Extremal statistics of random matrices, through the largest and smallest gaps, and extremes of characteristic polynomials.(5) A study of non-Hermitian random matrix theory, with connections between 2D Coulomb gases and the Gaussian free field.In order to understand Wigner's vision, in their recent proofs of fixed energy universality and eigenvector universality the investigator and collaborators developed new tools of interest for the above projects. These include random walks in dynamic random environments, coupling methods, and homogenization theory for partial differential equations with time-dependent random coefficients.