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

这个项目旨在解决概率论和统计物理交界处两个活跃而相关的领域，即大偏差理论(LDT)和极值理论的几个问题。LDT致力于估计罕见事件的概率，并了解这些事件发生的机制。该项目的主要关注点之一是随机网络中的罕见事件。随机网络是节点(或个体)的大型集合，其中节点对随机连接。该项目旨在用一些小规模模式的非典型数量的实例来描述随机网络的大规模结构，例如三个共同的朋友。这样的理解将对大型社交网络结构的统计估计产生影响。该项目的第二个目标是关于对数关联场的极值，它出现在从解析数论到数学生态学的一系列问题中。该项目的目的是在随机矩阵和反应扩散系统的背景下，促进对气候变化框架的普遍性和非普遍性方面的理解。该项目为研究生和本科生提供研究培训机会。关于LDT的问题集中在随机超图和随机矩阵的非线性函数的问题上。该项目将进一步发展一种基于张量分解的随机超图LDT的最新方法，与极值图论中的正则性方法和统计物理中的平均场近似相联系，并应用于研究用于建模社会网络的Gibbs度量。在随机矩阵方面，该项目将进一步推进最近通过球面积分分析极值特征值大偏差的方法，以便处理具有一般入口分布的模型，包括稀疏模型。该项目旨在开发灵活的工具来研究随机矩阵理论中的一大类模型，到目前为止，最有力的结果仅限于具有光滑对称性的经典系综。在不同的方向上，该项目将把概率方法扩展到反应-扩散方程的研究，以便研究具有边界相互作用的高维偏微分方程组，特别关注用于对入侵物种的传播进行建模的系统。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.