The primary focus of the project will be the study of random processes random walks and branching random walks, contact processes, Ising model, and competition processes on hyperbolic groups and graphs. The last four of these processes exhibit multiple phases, including an intermediate phase of weak survival that is not seen on integer lattices. A major objective of the research will be to understand how hyperbolic geometry constrains the upper phase transition between weak and strong survival, and how this in turn is related to asymptotic behavior of the Green's functions of random walk. This will involve development of techniques centered around Ancona inequalities, symbolic dynamics and infinite-dimensional Perron-Frobenius theory, and the Lyapunov-Schmidt reduction of infinite systems of algebraic functional equations. Secondary foci of the project will be (a) mathematical problems arising in Bayesian nonparametric function estimation, and (b) stochastic models of epidemics and interspecies competition.
Stochastic interacting particle systems are widely used as models in various areas of science, especially in statistical physics and in population biology, but also in the social sciences. It is hoped that studying such processes in infinite hyperbolic geometries will ultimately contribute to understanding their behavior in finite "expander" and "small-worlds" networks, which may better model interactions in various social and ecological systems.
Intellectual Merit This project focused mainly on mathematical models of epidemics, with emphasis on models attempting to account for geographical factors restricting the spread of infection. In these models, the population through which the epidemic spreads is distributed in "colonies" located in space. (For example, a one-dimensional space might be suitable for describing animal or plant populations that live along riverbeds or coastlines, while two-dimensional space would be better suited to species that live and interact in a grassland.) Infected individuals can infect only susceptible, non-infected individuals in neighboring colonies. These infections occur at random times, but only before the infected individual recovers. In some models, individuals acquire immunity from further infection upon recovery; in other models they rejoin the pool of susceptible individuals. In the former case, the susceptible population is eventually depleted to the point where the epidemic can no longer sustain itself locally, and so the epidemic proceeds in a fashion somewhat like a forest fire, staying alive only at the edges of its extent. The first figure attached illustrates a simulation of a very simple super-critical one-dimensional case, showing the local density of infected individuals in space and time, and clearly illustrating how the infection front travels at a definite speed, even though the epidemic is spread purely by random contacts. The second figure shows the progress of a critical epidemic. Our primary interest was in discovering how the spread of epidemics in space and time would be affected by changes in the parameters of the model, such as the infection and recovery rates and the population density. We were especially interested in how the epidemic would behave at the "critical values" of the parameters, for instance, when the infection rate is just high enough so that, in a colony where nearly all individuals are susceptible, an infected individual would on average infect one other before recovering. We found that here the spacial extent of the epidemic has a "long tail", that is, exhibits a very large range of variation from one run to the next, and we were able to formulate precise rules describing this variation as functions of the population density and infection/recovery rates. Broader Impacts (1) Stochastic processes of the type considered in this proposal have become increasingly important in population biology, epidemiology, and the emerging field of ``complex networks''. (2) Much of the research was done in collaboration with several Ph. D. students at the university of Chicago who have now finished their degrees and moved on to academic positions at leading research universities.
