Markov Random Fields, Finitary Codings, Phase Transitions and Ergodic Theory

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

0103841 Steif The principal investigator will carry out research in probabability theory; more specifically in statistical physics, Markov random fields and ergodic theory, and especially on problems which belong to the interface of these disciplines. A common theme for these problems is the notion of a phase transition which comes up in various guises. The first set of problems to be investigated concerns elucidating the relationship among the notion of finitary mappings, a concept which is relatively old, the much newer and important notion of exact simulation and finally the existence of a phase transition. It is already known that there are intimate connections between these. A second set of problems concerns better understanding the relationship between the phase diagram for various statistical mechanical systems (such as the Ising or the classical Heisenberg models) and the underlying graph on which they live. These questions lead to the study of phenomena which do not arise on the classical Euclidean lattices. A third set of problems concerns determining the sets of capacity 0 for certain concrete Markov processes arising in particle systems (e.g., the stochastic Ising model). Finally, a fourth set of problems concerns understanding the class of stationary processes which arise from positive contractions in Hilbert space, a class of stationary processes which have very different behavior than processes traditionally studied. Even for these systems, the concept of phase transition arises. Many systems in the world evolve stochastically and probability theory is utilized in order to better understand and predict the behavior of such systems. For example, statistical physics is the study of how particles behave globally and more generally how various general cooperative systems evolve. Understanding such systems is part of the motivation for this project. Another goal of this project is to determine how certain abrupt changes occur within a system (a so-called phase transition such as when water boils) and to investigate the underlying conditions which lead to such phenomena. There are various distributions which arise here which are important to investigate but for which one is not able to make explicit calculations. In such cases, a key tool is computers which, via simulations, allows us to approximate these distributions. In many cases, one can sample exactly from these distributions and one more goal of this project is to understand when this is possible.