Electronic Journal of Probability

Ergodic theory on stationary random graphs

Itai Benjamini and Nicolas Curien

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Stationary random graph Simple random walk Ergodic Theory Entropy Liouville Property

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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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