Mathematical Sciences: Percolation and Related Processes

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The principal investigator will conduct research concerning percolation theory, the random cluster model, and their relation to Ising and Potts models, mainly in two dimensions. Results to be sought concern (1) the formation of large rings of dual bonds in the random cluster model, which correspond to droplets of one phase immersed in another in a special construction of the Ising or Potts model; (2) the curve describing the typical shape of these rings, and its relation to properties of the function giving exponential rates of decay of correlations along rays emanating in various directions from the origin; and (3) percolation in half spaces. The principal investigator is working in the field of percolation theory and self avoiding random walk theory. The problems he will study are motivated by physical considerations. For example, percolation theory was motivated by the flow of a liquid through a porous material. Self avoiding random walks are used to model chemical polymers.