Rare Events and High-Dimensional Stochastic Systems

Brown University, Providence RI

Investigators

Abstract

Large collections of interacting random elements arise in many areas, ranging from physics and neuroscience to engineering and operations research. It is of great importance to study fluctuations and large deviations from the typical or mean behavior of these systems. Indeed, fluctuations and atypical events, although rare, can have significant impact, so it is important to quantify these probabilities and to understand typical occurrences of rare events. A significant mathematical challenge is to see how the structure of interaction between large collections of stochastic elements influences the nature of such deviations. This project will address this challenge for three classes of stochastic systems. The first class consists of large collections of interacting diffusions that arise as models of stock prices in finance, as continuum models of population dynamics in biology, and in statistical physics. The second class concerns high-dimensional measures, such as random ensembles of matrices arising as representations of high-dimensional data, and their relation to lower-dimensional projections, which are used as a dimension-reduction technique when analyzing data. Understanding the statistics and deviations of lower-dimensional projections is not only relevant for statistics and data science but also has significance for some open conjectures in convex geometry. The third class pertains to the study of fluctuations of eigenvectors of random matrices and addresses hypotheses related to quantum mechanical systems. The project will include vertically integrated mentoring of junior researchers at multiple levels and outreach efforts to foster broadening the mathematical participation of underrepresented and disadvantaged groups. This project will study high-dimensional stochastic systems and work to characterize fluctuations and large deviations from mean behavior and the nature of rare events in such systems. Three classes of problems will be considered. The first focuses on large collections of diffusions whose local interaction structure is governed by an underlying graph, and aims to study their atypical or large deviation behavior. While this has been well understood for almost half a century in the case when the underlying graph is the complete graph, the goal of this project is to study the complementary case when the graphs are (uniformly) sparse, which is also the relevant regime for many applications. This will require a combination of tools from random graph theory, stochastic analysis and variational methods. The second class of problems relates to the study of concentration and large deviation behavior of projections of high-dimensional convex bodies, with a focus on non-commutative settings, such as the level sets of norms of Banach spaces of matrices such as the p-Schatten spaces. The relation to some outstanding conjectures in convex geometry will also be explored. The third theme concerns the study of universality of the Eigenstate Thermalisation Hypothesis from physics, which is a statement about eigenvectors of random matrices, and corresponding fluctuations. These problems address fundamental problems in probability theory and have applications to statistical physics, asymptotic convex geometry and statistics. The project will employ a combination of analytical, geometric and probabilistic methods. 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.