Mathematical Sciences: Studies in Random Walk

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Lawler will investigate a number of questions about the geometric properties of simple random walk and harmonic measure. Included in the research is an analysis of the intersection exponent and resistance exponents, which are analogous to critical exponents in mathematical physics. Diffusion limited aggregation is a cluster model based on random walk dynamics. The main goal will be to prove theorems about the growth rate, shape, and convergence rate for these clusters. Random walk models are used in a number of areas of physics and other natural sciences. The investigator will continue research to understand rigorously the mathematical properties of some of these models. Of particular interest are problems on the intersection of random walks and walks with constraints which arise in statistical mechanics and critical phenomenon. Another area is cluster formation, especially clusters which grow using random walk dynamics. These models arise in the study of dendritic growth and in the study of interfaces.