Random interlacements and amenability

TL;DR

This paper characterizes the connectivity properties of random interlacements on transitive graphs, showing that connectivity occurs if and only if the graph is amenable, and explores phase transitions in nonamenable cases.

Contribution

It proves the equivalence between amenability and connectivity of interlacement sets on transitive graphs, and analyzes phase transitions in nonamenable graphs.

Findings

Connectivity of interlacement set iff the graph is amenable.

In nonamenable graphs, small u yields infinitely many infinite clusters.

For some nonamenable graphs, large u results in a connected interlacement set.

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.