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

Duke University, Durham NC

Investigators

Abstract

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.