Mathematical Sciences: Random Trees and Tree-Indexed Processes

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Tree-indexed processes are a way of studying classical tree-related probability models in a unified and more powerful way. Examples of such models are: branching random walk, branching Markov chains, first passage percolation on a tree, parallel searches, random and non-random Cantor sets and hyperbolic manifolds. Current work on tree-indexed processes is focused on extending some results on Hausdorff dimensions of random sets and on determining exact capacity criteria for subsets of the range space to be polar for a given tree-indexed process. Probability models with branching or tree structure arise in many areas of science. Much of the classical work on branching processes was motivated by random family trees that arise in population genetics; the theory of cascades encompasses atomic chain reactions and rainfall models; more recently, random trees have been used to study self-similarity (fractal behavior) in spatial models. By abstracting the common structure of these models into "tree-indexed processes", we have been able to prove many new results about this class of models, as well as strengthening previously known results.