Analytic Methods for the Random Matrix Universality Class

随机矩阵普适性类的解析方法

• 批准号: 1513587
• 负责人: Paul Bourgade
• 金额: $ 22.5万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1513587&HistoricalAwards=false
• 关键词: Analytic; Methods; Random; Matrix; Universality

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.

研究人员将研究随机矩阵，这一领域起源于尤金·维格纳的想法，即随机矩阵模拟物理系统的光谱行为。多年来，随机矩阵的应用超越了纯粹的数学，并被用于统计学、计算机科学、电信以及更广泛的信息网络中。这项研究项目的重点是普遍性--即极限光谱统计只取决于对称类型而不取决于底层系统的其他细节的现象。现在，随机矩阵统计在可积系统、增长模型和数论的许多方面都有应用。在过去的几年里，已经取得了深刻的进展，并为越来越多的随机矩阵建立了普适性。本课题的研究内容包括：(1)随机带矩阵的普适性和量子唯一遍历性。其动机是试图通过这些随机薛定谔算子的玩具模型来接近Anderson相变。(2)通过最近引入的一种新的随机游动--特征向量矩流，对非微扰区域的本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贝塔系综本征值的个别波动。(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，与二维库仑气体和高斯自由场之间的联系。为了理解Wigner的愿景，在他们最近关于固定能量普适性和本征向量普适性的证明中，研究人员和合作者为上述项目开发了感兴趣的新工具。其中包括动态随机环境中的随机游动、耦合方法和具有随时间变化的随机系数的偏微分方程齐化理论。