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June 2013 Random interlacements and amenability
Augusto Teixeira, Johan Tykesson
Ann. Appl. Probab. 23(3): 923-956 (June 2013). DOI: 10.1214/12-AAP860

Abstract

We consider the model of random interlacements on transient graphs, which was first introduced by Sznitman [Ann. of Math. (2) (2010) 171 2039–2087] for the special case of ${\mathbb{Z}}^{d}$ (with $d\geq3$). In Sznitman [Ann. of Math. (2) (2010) 171 2039–2087], it was shown that on ${\mathbb{Z}}^{d}$: for any intensity $u>0$, the interlacement set is almost surely connected. The main result of this paper says that for transient, transitive graphs, the above property holds if and only if the graph is amenable. In particular, we show that in nonamenable transitive graphs, for small values of the intensity $u$ the interlacement set has infinitely many infinite clusters. We also provide examples of nonamenable transitive graphs, for which the interlacement set becomes connected for large values of $u$. Finally, we establish the monotonicity of the transition between the “disconnected” and the “connected” phases, providing the uniqueness of the critical value $u_{c}$ where this transition occurs.

Citation

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Augusto Teixeira. Johan Tykesson. "Random interlacements and amenability." Ann. Appl. Probab. 23 (3) 923 - 956, June 2013. https://doi.org/10.1214/12-AAP860

Information

Published: June 2013
First available in Project Euclid: 7 March 2013

zbMATH: 1375.60139
MathSciNet: MR3076674
Digital Object Identifier: 10.1214/12-AAP860

Subjects:
Primary: 60K35 , 82C41

Keywords: amenability , Graphs , Random interlacements , Random walks

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.23 • No. 3 • June 2013
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