Electronic Journal of Probability

Ergodic theory on stationary random graphs

Itai Benjamini and Nicolas Curien

Full-text: Open access

Abstract

A stationary random graph is a random rooted graph whose distribution is invariant under re-rooting along the simple random walk. We adapt the entropy technique developed for Cayley graphs and show in particular that stationary random graphs of subexponential growth are almost surely Liouville, that is, admit no non constant bounded harmonic functions. Applications include the uniform infinite planar quadrangulation and  long-range percolation clusters.

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 93, 20 pp.

Dates
Accepted: 29 October 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062415

Digital Object Identifier
doi:10.1214/EJP.v17-2401

Mathematical Reviews number (MathSciNet)
MR2994841

Zentralblatt MATH identifier
1278.05222

Subjects
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 28D20: Entropy and other invariants

Keywords
Stationary random graph Simple random walk Ergodic Theory Entropy Liouville Property

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Benjamini, Itai; Curien, Nicolas. Ergodic theory on stationary random graphs. Electron. J. Probab. 17 (2012), paper no. 93, 20 pp. doi:10.1214/EJP.v17-2401. https://projecteuclid.org/euclid.ejp/1465062415


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