Questions at the Interface of Probability and Geometry

University Of Chicago, Chicago IL

Investigators

Abstract

A common thread in the mathematical description of dynamical processes is the transport of some important quantity -- perhaps electrons in a crystal with imperfections, or infection in a biological network, or information in a social network -- by rules that incorporate randomness at the microscopic level, but result in statistical regularity at macroscopic scales. In many such processes, the key to understanding this macroscopic order is a careful analysis of the random motions of individual particles (or bits of information) in the network. In turn, understanding the random motions of particles in inhomogeneous networks often requires analysis of the corresponding motions in homogeneous networks. A major goal of this research is to construct a comprehensive picture of how random walkers behave in infinite homogeneous networks with tree-like, or "hyperbolic'' geometry. A secondary goal is to determine the behavior of a number of specific models of epidemic and information-propagation models in such networks. Some of the research topics will involve graduate students, who will benefit from exposure to these cutting-edge problems in mathematical probability theory and their motivating application areas. The project aims to settle a number of important open technical questions concerning the long-time behavior of random walks on lattices of semi-simple Lie groups and other nonamenable discrete groups. These center on "local limit" behavior: for which groups do random walks with arbitrary, but finitely supported, step distributions obey a universal local limit law? Related questions concern the structure of the Martin boundaries for such random walks, the support of the exit measure, and the geometry of long random loops. The project will also devote some effort to the study of SIR epidemics, with the particular object of describing critical scaling behavior.