Decoupling inequalities and interlacement percolation on <Emphasis Type="Italic">G</Emphasis>×ℤ |
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Authors: | Alain-Sol Sznitman |
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Institution: | (1) Department of Anatomy, Physiology and Pharmacology, College of Veterinary Medicine, Auburn University, 109 Greene Hall, Auburn, AL 36849, USA |
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Abstract: | We study the percolative properties of random interlacements on G×ℤ, where G is a weighted graph satisfying certain sub-Gaussian estimates attached to the parameters α>1 and 2≤β≤α+1, describing the respective polynomial growths of the volume on G and of the time needed by the walk on G to move to a distance. We develop decoupling inequalities, which are a key tool in showing that the critical level u
∗ for the percolation of the vacant set of random interlacements is always finite in our set-up, and that it is positive when
α≥1+β/2. We also obtain several stretched exponential controls both in the percolative and non-percolative phases of the model.
Even in the case where G=ℤ
d
, d≥2, several of these results are new. |
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Keywords: | |
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