Interacting Particle Systems

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0301795Liggett The project addresses several problems in interacting particle systems and related areas of probability theory. One collection of problems involves the stationary distributions for exclusion processes with drift in both one and higher dimensions. Recent work of the investigator showed the existence of stationary blocking measures in one dimension under weak assumptions. The next set of problems concerns the existence of nonblocking stationary profile measures in one dimension and of nonreversible stationary blocking measures in higher dimensions -- in neither case are there any known examples. Other problems involve (a) connections between the symmetric exclusion process and negative dependence, (b) the extent to which the known theory of the symmetric exclusion process extends to processes with a few asymmetries, (c) the analysis of some infinite systems motivated by the investigator's recent work on a class of models in sociology, and (d) the computation of the critical values for a family of reversible growth models on homogeneous trees. Many problems in the natural and social sciences involve the behavior of large numbers of individuals (people, molecules, cars, viruses, etc.) that interact in unpredictable ways. This lack of predictability is modeled by randomness. One of the fundamental contributions of probability theory in the past century is the realization that large populations display a significant amount of predictability and order in spite of the lack of predictability of the actions of the individuals in the population. This project is aimed at gaining a better understanding of several models of this general type. Among them are: (a) the exclusion process, which models particle motion, traffic flow, and the behavior of ribosomes in biology, (b) certain growth models that are partly motivated by tumor growth and conflicts between competing populations, and (c) models from sociology, in which relationships among individuals are altered by the exchange of gifts or rewards. In each case, the objective is to determine the long time behavior of the system.