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

   
@article{EJP2122,
	author = {Balázs Ráth and Artëm Sapozhnikov},
	title = {The effect of small quenched noise on connectivity properties of random interlacements},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {18},
	year = {2013},
	keywords = {Random interlacements; Bernoulli percolation; slab; vacant set; quenched noise; long-range correlations; transience},
	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 $\varepsilon\to 0$.
},
	pages = {no. 4, 1-20},
	issn = {1083-6489},
	doi = {10.1214/EJP.v18-2122},    
        url = {http://ejp.ejpecp.org/article/view/2122}}