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Random walks and probabilities of genetic identity in a tree

The purpose here is to discuss isotropic random walks on a set of graphs called infinite homogeneous trees, and to describe a population genetics model on these graphs such that migration between generations is represented by an isotropic random walk. The population model is of “Stepping Stone” type, which are used in Biology to model selectively neutral gene flow in a stationary interbreeding population. Here the main influences are local migration and the consolidative effects of random finite offspring distributions. In one or two dimensions, these random effects cause expanding waves which can mimic the “waves of advance” of epidemiological models or genetic models with selection. In three or more dimensions, or in infinite trees, there are still waves of advance but they do not exclude other types behind the “wave” (see §3 below).