Random interlacements and amenability

Details Random interlacements and amenability Random interlacements and amenability

2011-03-10

arxiv.org/abs/1103.2109v1

Mathematics

Probability

28 pages, 2 figures

Scientific paper

We consider the model of random interlacements on transient graphs, which was first introduced by A-S. Sznitman in arXiv:0704.2560 for the special case of Z^d (with d > 2). There it was shown that on 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 non-amenable transitive graphs, for small values of the intensity u, the interlacement set has infinitely many infinite clusters. We also provide examples of non-amenable 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.

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