Uniqueness of Gibbs Measures and Rapidly Mixing Dynamics

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9800351TetaliThis research concerns the hard-core lattice gas model on the integer lattice, with special emphasis on the square lattice. The project includes the following problems:(i) To establish a rigorous, quantitative connection between the rate of approach to equilibrium of certain discrete dynamics and the uniqueness of (infinite--volume) Gibbs measure for the hard-core model on general graphs(ii) To analyze the rate of mixing of certain dynamics with the intention of improving the existing bounds on the critical activity of the hard-core model on the square lattice.(iii) To investigate the role of percolation and other techniques in this context. In graph-theoretic terms, one of the objectives is to generate independent sets of a large n by n grid with probabilities favoring the bigger independent sets, in particular, with hard-core distribution with activity 1. The proposal also includes a student project on estimating the mixing rates of Markov chains on combinatorial structures, whose counting function is the nth Catalan number.The lattice gas model is of independent interest and significance to researchers in statistical physics, theory of computing, probabilistic combinatorics, and communication networks. The easiest-to-describe motivation is that of a telephone (loss) network with Poisson arrival rates and independent holding times for calls. The hard constraint demands that a call be accepted/active only if all the neighboring cells (neighborhood according to an arbitrary topology) are inactive. The PI hopes to contribute to the understanding of the connections between spatial correlations (i.e. the effect of the presence of a call in one location on calls at reasonably far locations) and the convergence of the dynamics (i.e. the rate at which the network approaches, if it eventually does, a unique stable configuration of active calls).