Ergodic Theory and Interacting Particle Systems

University Of Washington, Seattle WA

Investigators

Abstract

Abstract There is a long and fruitful history of interaction between various branches of analysis and probability theory. This proposal focuses furthering these connections, both by applying recent results in probability to analyze dynamical systems, and using ergodic theory and other branches of analysis, to study questions in probability. Two probabilistic models which have seen large contributions from ergodic theory are first passage percolation and Richardson's growth model. The work in this proposal aims to apply techniques from ergodic theory to study questions of coexistence of multiple infections in Richardson's growth models and the existence of one sided geodesics in first passage percolation. One proposed application of probability results to study dynamical systems is the use of domino tilings the possible ergodic properties of two dimensional subshifts of finite type. Another is to use recent advances in necessary conditions for ergodicity of g measures to answer a longstanding open question about the necessary conditions modulus of continuity of an Anosov map on the torus which ensures ergodicity of the corresponding measure preserving system. Mathematicians have long used probabilistic models to gain understanding of complex physical, biological and social systems. Examples of these models include the Ising model to understand the process of magnetization, random graphs to study the spread of disease through sexual contact, and Richardson's growth model to understand the spread of infection. This proposal aims to contribute to this field by using new techniques in the mathematical field of dynamical systems to analyze probabilistic models inspired by physics and biology.