We study transience and recurrence of simple random walks on percolation clusters in the hierarchical group of order N, which is an ultrametric space. The connection probability on the hierarchical group for two points separated by distance k is of the form (Formula presented.), with (Formula presented.), non-negative constants (Formula presented.), and (Formula presented.). Percolation occurs for (Formula presented.), and for the critical case, (Formula presented.), (Formula presented.) and sufficiently large (Formula presented.). We show that in the case (Formula presented.) the walk is transient, and in the case (Formula presented.) there exists a critical (Formula presented.) such that the walk is recurrent for (Formula presented.) and transient for (Formula presented.). The proofs involve ultrametric random graphs, graph diameters, path lengths, and electric circuit theory. Some comparisons are made with behaviours of simple random walks on long-range percolation clusters in the one-dimensional Euclidean lattice.

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Keywords Hierarchical group, Percolation, Random graph, Random walk, Recurrence, Renormalization, Transience, Ultrametric space
Persistent URL dx.doi.org/10.1007/s10959-016-0691-7
Journal Journal of Theoretical Probability
Dawson, D.A, & Gorostiza, L.G. (2016). Transience and Recurrence of Random Walks on Percolation Clusters in an Ultrametric Space. Journal of Theoretical Probability, 1–33. doi:10.1007/s10959-016-0691-7