## Electronic Journal of Probability

### Ergodic theory on stationary random graphs

#### 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

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
Secondary: 28D20: Entropy and other invariants

Rights

#### 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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