Percolation theory and spatial interaction

Research Project, 2012 – 2015

Percolation theory is one of the most lively areas in probability theory, and is of considerable importance to the grand project of statistical mechanics, which is to understand how microscopic laws give rise to macroscopic phenomena. The purpose of this project is to contribute to percolation theory and the theory of interacting spatial stochastic systems. Specific goals include (a) proving (or possibly disproving) a conjectured characterization of amenability of transitive graphs in terms of uniqueness in percolation models, (b) proving (or possibly disproving) a highly intuitive monotonicity property of biased random walk on percolation clusters, (c) developing a disagreement percolation technique for analysing phase transition and spatial correlations in interacting point processes, (d) exploring how far the correlation inequality known as the FKG inequality holds in the Potts model, and (e) proving (or possibly disproving) that symmetric growth rates is a necessary condition for mutual unbounded growth in the two-type Richardson model in two or more dimensions. It is unlikely that all of these problems will be solved within the time span of the project, since especially problems (a), (b) and (e) have already to varying extent demonstrated resistance against early attempts to solve them, but significant progress on most of them can be expected.

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