Entropy of Random Walk Range on Uniformly Transient and on Uniformly Recurrent Graphs

David Windisch (The Weizmann Institute of Science)

Abstract


We study the entropy of the distribution of the set $R_n$ of vertices visited by a simple random walk on a graph with bounded degrees in its first n steps. It is shown that this quantity grows linearly in the expected size of $R_n$ if the graph is uniformly transient, and sublinearly in the expected size if the graph is uniformly recurrent with subexponential volume growth. This in particular answers a question asked by Benjamini, Kozma, Yadin and Yehudayoff (preprint).

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Pages: 1143-1160

Publication Date: July 7, 2010

DOI: 10.1214/EJP.v15-787

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