Random Processes on Graphs, and Planar Brownian Motion

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9803597 Peres The principal investigator will analyze various random processes on graphs. For instance, a model of noisy propagation on trees has been studied independently in Statistical Mechanics, in Computer Science and in Genetic Reconstruction. This model is well understood only when the propagating variables take two values, when it can be identified with the Ising model. There are two distinct critical temperatures, one for uniqueness of Gibbs states and another for purity of the Markovian state that corresponds to free boundary conditions. The second critical temperature is not known for the Potts model, where the propagating variables can take more than two values. To study large homogeneous graphs that are not trees, the investigator will apply a new "mass transport method" developed recently with several collaborators. Problems on the geometry of the planar Brownian path will also be investigated; a key tool will be the intersection equivalence of the paths with random sets constructed via "fractal percolation". Mathematical biologists studying the spread of genetic mutations in a family tree, and computer scientists studying the propagation of errors in a noisy communication network, were led independently to the same mathematical model that was considered earlier in statistical mechanics. The principal investigator will attempt to unify and extend some of the earlier results on these models, by using recent probabilistic techniques; some of this involves collaboration with computer scientists. Recently, new methods have emerged to exploit symmetries in the geometry of the underlying network, to obtain better understanding of random processes on it; the investigator will develop and apply these methods further. Finally, the trace of the erratic random motion of a particle has been studied intensively by probabilists; this research will investigate certain new approaches to these traces to try and elucidate their geometric structure.