Transience and Recurrence of Random Walks on Percolation Clusters in an Ultrametric Space

D. A. Dawson () and L. G. Gorostiza
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D. A. Dawson: Carleton University
L. G. Gorostiza: CINVESTAV

Journal of Theoretical Probability, 2018, vol. 31, issue 1, 494-526

Abstract: Abstract 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 $$c_k/N^{k(1+\delta )}, \delta >0$$ c k / N k ( 1 + δ ) , δ > 0 , with $$c_k=C_0+C_1\log k+C_2k^\alpha $$ c k = C 0 + C 1 log k + C 2 k α , non-negative constants $$C_0, C_1, C_2$$ C 0 , C 1 , C 2 , and $$\alpha >0$$ α > 0 . Percolation occurs for $$\delta 0$$ α > 0 and sufficiently large $$C_2$$ C 2 . We show that in the case $$\delta 0,\alpha >0$$ δ = 1 , C 2 > 0 , α > 0 there exists a critical $$\alpha _\mathrm{c}\in (0,\infty )$$ α c ∈ ( 0 , ∞ ) such that the walk is recurrent for $$\alpha \alpha _\mathrm{c}$$ α > α c . 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.

Keywords: Percolation; Hierarchical group; Ultrametric space; Random graph; Renormalization; Random walk; Transience; Recurrence; Primary 05C80; 05C81; 60K35; Secondary 60C05 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-016-0691-7

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