Large Deviations and Extremes for Random Matrices, Tensors, and Fields

随机矩阵、张量和场的大偏差和极值

• 批准号: 2154029
• 负责人: Nicholas Cook
• 金额: $ 26万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154029&HistoricalAwards=false
• 关键词: Large; Deviations; Extremes; Random; Matrices

This project aims to address several problems in two active and related areas at the interface of probability theory and statistical physics, namely, large deviations theory (LDT) and extreme value theory. LDT is concerned with estimating the probabilities of rare events, and understanding the mechanisms by which these events arise. One of the main focuses of the project is on rare events for random networks. Random networks are large collections of nodes (or individuals) where pairs of nodes are connected at random. The project aims to describe the large-scale structure of random networks with an atypical number of instances of some small-scale pattern, such as three mutual friends. Such an understanding would have implications for statistical estimation of the structure of large social networks. The second aim of the project concerns extreme values for logarithmically correlated fields (LCFs), which arise in problems ranging from analytic number theory to mathematical ecology. The project aims to advance the understanding of universal and non-universal aspects of LCFs in the context of random matrices and reaction-diffusion systems. The project provides research training opportunities for graduate and undergraduate students. The problems concerning LDT focus on questions about nonlinear functions of random hypergraphs and random matrices. The project will further develop a recent approach to LDT for random hypergraphs based on tensor decompositions, with connections to the regularity method in extremal graph theory and the mean-field approximation in statistical physics, and with applications to the study of Gibbs measures used to model social networks. In the context of random matrices, the project will further advance a recent approach to large deviations of extremal eigenvalues through the analysis of spherical integrals, in order to address models with general entry distributions, including sparse models. The project on extreme values for LCFs aims to develop flexible tools to study a broad class of models in random matrix theory, where the strongest results to date are confined to classical ensembles with smooth symmetries. In a different direction, the project will extend a probabilistic approach to the study of reaction-diffusion equations in order to study coupled systems of partial differential equations in higher dimensions with boundary interactions, with particular attention to systems used to model the propagation of invasive species.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.

该项目的目的是解决概率理论和统计物理界面的两个活跃和相关领域的几个问题，即大偏差理论（LDT）和极值理论。 LDT关注的是估计罕见事件的概率，并了解这些事件出现的机制。该项目的主要重点之一是用于随机网络的罕见事件。随机网络是大量的节点（或个体）集合，其中随机连接了一对节点。该项目旨在用一些小规模模式（例如三个共同的朋友）的非典型实例来描述随机网络的大规模结构。这种理解将对大型社交网络结构的统计估计有影响。该项目的第二个目的涉及对数相关场（LCF）的极端价值，这是在从分析数理论到数学生态学的问题中出现的。该项目旨在在随机矩阵和反应扩散系统的背景下促进对LCF的普遍和非普遍方面的理解。该项目为研究生和本科生提供了研究培训机会。 有关LDT的问题集中在有关随机超图和随机矩阵的非线性功能的问题上。该项目将进一步开发一种基于张量分解的随机超图的最新方法，并在极端图理论中与规则性方法的连接以及统计物理学中的均值场近似以及用于模拟社交网络的GIBBS测量的应用。在随机矩阵的背景下，该项目将通过分析球形积分来进一步推进最新的极端特征值偏差的方法，以解决具有一般进入分布（包括稀疏模型）的模型。 LCF的极端价值项目旨在开发灵活的工具来研究随机矩阵理论中的一系列模型，迄今为止，最强的结果仅限于具有光滑对称性的经典合奏。在不同的方向上，该项目将向反应扩散方程的研究扩展一种概率方法，以研究较高维度和边界相互作用的偏微分方程的耦合系统，特别关注用于建模入侵物种的系统。该奖项通过评估范围进行了评估，该奖项反映了众所周知的范围，以反映了该奖项的范围。