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2012 Ergodic theory on stationary random graphs
Itai Benjamini, Nicolas Curien
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Electron. J. Probab. 17: 1-20 (2012). DOI: 10.1214/EJP.v17-2401


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.


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Itai Benjamini. Nicolas Curien. "Ergodic theory on stationary random graphs." Electron. J. Probab. 17 1 - 20, 2012.


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

zbMATH: 1278.05222
MathSciNet: MR2994841
Digital Object Identifier: 10.1214/EJP.v17-2401

Primary: 05C80
Secondary: 28D20

Keywords: Entropy , ergodic theory , Liouville property , Simple random walk , Stationary random graph

Vol.17 • 2012
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