Phase Transition: Questions in Percolation and Interacting Particle Systems

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9704197 Liggett Alan Stacey proposes to work on problems in percolation, interacting particle systems and probabilistic combinatorics. He proposes to work (partly with Tom Liggett) to try to extend work of theirs (and others) on contact processes on trees: to prove a conjecture of Liggett which would complete a significant characterization of the three phases of the process on a homogeneous tree; to understand more fully the behavior in the three phases; and to use this detailed knowledge of the behavior to complete work with Robin Pemantle in which certain exponentially growing non-homogeneous trees are found where the branching random walk and/or the contact process only have one phase transition. Alan Stacey also proposes to extend methods developed by van den Berg to show almost sure uniqueness of the Gibbs measure of a two-dimensional spin-glass model at parameter values where the Gibbs measure for the corresponding Ising model is not unique. He proposes to work on problems in percolation such as trying to prove that percolation occurs in certain models where this is not known for any non-trivial value of the parameters, including a percolation model with long-range dependencies which arises from a question about random walks on a graph; a more ambitious project (in the light of the unsuccessful attempts by many distinguished probabilists) is to show that the percolation probability at the critical value is zero for standard bond percolation in three dimensions. Stacey also proposes to work on an appealing conjecture that the rates of convergence are the same for two different Markov chains arising from a card-shuffling situation. The problems described concern mathematical models for important physical and biological processes such as the spread of disease and the growth of populations (both human, and those consisting of other organisms). Although the mathematical descriptions are abstract and are not typically based on one specific real-world situation, a detailed understandi ng of the abstract models helps one to understand and predict important physical model; indeed, one or two simply described mathematical models can be of relevance to a very large number of physical systems. One key phenomenon common to the abstract models and the empirical observations is phase transitions: a system can exhibit two or more quite different types of behavior (or phases) under different conditions; a classic example is that of a disease which may spread out of control under certain conditions and fizzle out under different circumstances. The change from one phase to another is often quite dramatic and can be the result of a very small change in the parameters on which the system depends, such as the susceptibility of the individuals in a population or (in another classic example of phase transition -- that of a substance turning from a solid to a liquid) the temperature. The work proposed explores the nature of the phase transition and the types of phases that can occur in a number of different models. They include the contact process -- a model for the spread of disease -- which can exhibit two phases in certain environments, and three phases in others; and spin glasses -- which model magnetism in a substance with impurities -- where, in a certain case, it is thought that no phase transition exists and the system has similar behavior at all temperatures; but this conjecture is not yet proven.