CAREER: Distances in Random Media

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

The projects to be supported by this CAREER award center on analyzing distances in random environments. Some examples include finding travel times on road systems with random traffic, the minimal number of connections between people in social media graphs, and generally the total cost of moving from one place to another in a medium where microscopic distances are unknown. The research questions are combined with educational activities, like summer workshops on probability theory for young researchers, summer projects with undergraduates, and funding for a postdoctoral researcher. The proposed work has connections to other areas of mathematical physics, like the structure of disordered magnets, and satisfaction problems from computer science. This proposal details projects in probability theory and mathematical physics, specifically in statistical mechanics. The main models are percolation models and first-passage percolation (FPP), and the questions are to analyze the extreme values of collections of random variables that are highly correlated. In FPP, the main observable is the travel time from one node of a network to another one in a random environment, and we plan to study the asymptotics and fluctuations of this travel time, along with the structure of minimizers (geodesics) and near-minimizers. Much of the research will involve Busemann functions, which are tools from metric geometry used to study parallelism of geodesic rays. It is expected that results obtained in these studies will have consequences in spatial growth and epidemic models, other percolation problems, disordered spin systems, and polymer models. The proposed work includes an educational component, with funds for a postdoctoral researcher, summer workshops, and summer projects for undergraduates.