## Electronic Journal of Probability

### 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 $\mathbb{Z}^d$ 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 $\mathbb{Z}^d$ 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 $\varepsilon$, 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 $\varepsilon$, independently of the randomness of the interlacement, and independently for different vertices. We prove that for any $d\geq 3$ and $u>0$, almost surely, the perturbed random interlacement percolates for small enough noise parameter $\varepsilon$. 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 $d\geq 3$, there is still a non trivial phase transition in $u$ when the noise parameter $\varepsilon$ is small enough, and we give explicit upper and lower bounds on the value of the critical threshold, when $\varepsilon\to 0$.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 4, 20 pp.

Dates
Accepted: 7 January 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064229

Digital Object Identifier
doi:10.1214/EJP.v18-2122

Mathematical Reviews number (MathSciNet)
MR3024098

Zentralblatt MATH identifier
1347.60132

Rights

#### Citation

Ráth, Balázs; Sapozhnikov, Artëm. The effect of small quenched noise on connectivity properties of random interlacements. Electron. J. Probab. 18 (2013), paper no. 4, 20 pp. doi:10.1214/EJP.v18-2122. https://projecteuclid.org/euclid.ejp/1465064229

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