Professor Yang will work on a number of problems in random fields and Markov processes with local interactions. In both cases, a question of primary interest is the investigation and description of critical phenomena. Professor Yang will study the critical dimensions of the scaling limits of self-avoiding random walks and the structure of infinite volume limits of Markov processes with local interactions. He will also investigate the structure of the Gibbs state of lattice gas models with hard- core conditions and the differentiability of pressures of the system, and will give a description of how a non-equilibrium distribution approaches equilibrium in such models. This project is research in a part of probability theory that is closely related to mathematical physics. The motivating physical problem is to predict the long-run behavior of systems of large numbers of interacting particles, based on information of a local nature about the interactions. Some of the techniques of quantum field theory for investigating such systems turn out to be applicable to the study of certain kinds of random processes that arise in probability theory, for instance random motion constrained by the condition that the path of the randomly moving point not cross itself.
