The effect of small quenched noise on connectivity properties of random interlacements

Abstract

Random interlacements (at level u) is a one parameter family of random subsets of Zd introduced by Sznitman. The vacant set at level u is the complement of the random interlacement at level u. While the random interlacement induces a connected subgraph of Zd for all levels u, the vacant set has a non-trivial phase transition in u . In this paper, we study the effect of small quenched noise on connectivity properties of the random interlacement and the vacant set. For a positive ε , we allow each vertex of the random interlacement (referred to as occupied) to become vacant, and each vertex of the vacant set to become occupied with probability ε, independently of the randomness of the interlacement, and independently for different vertices. We prove that for any d≥3 and u>0, almost surely, the perturbed random interlacement percolates for small enough noise parameter ε. In fact, we prove the stronger statement that Bernoulli percolation on the random interlacement graph has a non-trivial phase transition in wide enough slabs. As a byproduct, we show that any electric network with i.i.d. positive resistances on the interlacement graph is transient. As for the vacant set, we show that for any d≥3, there is still a non trivial phase transition in u when the noise parameter ε is small enough, and we give explicit upper and lower bounds on the value of the critical threshold, when ε→0.