The Annals of Applied Probability

Random interlacements and amenability

Augusto Teixeira and Johan Tykesson

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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.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 3 (2013), 923-956.

Dates
First available in Project Euclid: 7 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1362684850

Digital Object Identifier
doi:10.1214/12-AAP860

Mathematical Reviews number (MathSciNet)
MR3076674

Zentralblatt MATH identifier
1375.60139

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Random interlacements random walks graphs amenability

Citation

Teixeira, Augusto; Tykesson, Johan. Random interlacements and amenability. Ann. Appl. Probab. 23 (2013), no. 3, 923--956. doi:10.1214/12-AAP860. https://projecteuclid.org/euclid.aoap/1362684850


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