Electronic Journal of Probability

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

David Windisch

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60J05: Discrete-time Markov processes on general state spaces

random walk range entropy

This work is licensed under aCreative Commons Attribution 3.0 License.


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

Export citation


  • I. Benjamini, G. Kozma, A. Yadin, A. Yehudayoff. Entropy of random walk range. Ann. Inst. H. Poincaré Probab. Statist., to appear.
  • T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein. Introduction to algorithms. (second edition) MIT Press, Cambridge, MA, 2001.
  • T.M. Cover, J.A. Thomas. Elements of information theory. John Wiley & Sons, Inc., New York, 1991.
  • R. Durrett. Probability: Theory and Examples. (third edition) Brooks/Cole, Belmont, 2005.
  • A. Erschler. On drift and entropy growth for random walks on groups. Ann. Probab. 31/3, pp. 1193-1204, 2003.
  • V.A. Kaimanovich, A.M. Vershik. Random walks on discrete groups: boundary and entropy. Ann. Probab., 11/3, pp. 457-490, 1983.
  • A. Naor, Y. Peres. Embeddings of discrete groups and the speed of random walks. International Mathematics Research Notices 2008, Art. ID rnn 076.
  • P. Révész. Random walk in random and non-random environments. Second edition. World Scientific Publishing Co. Pte. Ltd., Singapore, 2005.
  • C.E. Shannon. A mathematical theory of communication. The Bell System Technical Journal, Vol. 27, pp. 379-423, 623-656, 1948.
  • N.T. Varopoulos. Long range estimates for Markov chains. Bull. Sci. Math. (2) 109, pp. 225-252, 1985.
  • W. Woess. Random walks on infinite graphs and groups. Cambridge University Press, Cambridge, 2000.