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

TL;DR

This paper investigates how small random perturbations affect the connectivity and phase transition properties of random interlacements and their vacant sets in high-dimensional integer lattices, revealing persistence of phase transitions under noise.

Contribution

It demonstrates that small quenched noise preserves phase transitions in the connectivity of random interlacements and vacant sets, extending understanding of their robustness.

Findings

Random interlacement percolates under small noise in all dimensions d>=3.

The vacant set maintains a non-trivial phase transition with small noise, with explicit bounds on critical thresholds.

Electric networks on the interlacement graph are shown to be transient, strengthening previous results.

Abstract

The random interlacements (at level u) is a one parameter family of random subsets of Z^d introduced by Sznitman in arXiv:0704.2560. The vacant set at level u is the complement of the random interlacement at level u. In this paper, we study the effect of small quenched noise on connectivity properties of the random interlacement and the vacant set. While the random interlacement induces a connected subgraph of Z^d for all levels u, the vacant set has a non-trivial phase transition in u, as shown in arXiv:0704.2560 and arXiv:0808.3344. For a positive epsilon, 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 epsilon, 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 epsilon. 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, which strengthens our result in arXiv:1102.4758. 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 epsilon is small enough, and we give explicit upper and lower bounds on the value of the critical threshold, when epsilon tends to 0.