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2013 The effect of small quenched noise on connectivity properties of random interlacements
Balázs Ráth, Artëm Sapozhnikov
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Electron. J. Probab. 18: 1-20 (2013). DOI: 10.1214/EJP.v18-2122

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 $\varepsilon\to 0$.

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Balázs Ráth. Artëm Sapozhnikov. "The effect of small quenched noise on connectivity properties of random interlacements." Electron. J. Probab. 18 1 - 20, 2013. https://doi.org/10.1214/EJP.v18-2122

Information

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

zbMATH: 1347.60132
MathSciNet: MR3024098
Digital Object Identifier: 10.1214/EJP.v18-2122

Subjects:
Primary: 60K35
Secondary: 82B43

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