## Electronic Journal of Probability

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

David Windisch

#### 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).

#### Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 36, 1143-1160.

Dates
Accepted: 7 July 2010
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819821

Digital Object Identifier
doi:10.1214/EJP.v15-787

Mathematical Reviews number (MathSciNet)
MR2659760

Zentralblatt MATH identifier
1226.60070

Keywords
random walk range entropy

Rights

#### Citation

Windisch, David. Entropy of Random Walk Range on Uniformly Transient and on Uniformly Recurrent Graphs. Electron. J. Probab. 15 (2010), paper no. 36, 1143--1160. doi:10.1214/EJP.v15-787. https://projecteuclid.org/euclid.ejp/1464819821

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