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A lower bound on the critical parameter of interlacement percolation in high dimension
, 2009
"... We investigate the percolative properties of the vacant set left by random interlacements on Z d, when d is large. A nonnegative parameter u controls the density of random interlacements on Z d. It is known from [15], [14], that there is a nondegenerate critical value u∗, such that the vacant set a ..."
Abstract

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We investigate the percolative properties of the vacant set left by random interlacements on Z d, when d is large. A nonnegative parameter u controls the density of random interlacements on Z d. It is known from [15], [14], that there is a nondegenerate critical value u∗, such that the vacant set at level u percolates when u < u∗, and does not percolate when u> u∗. Little is known about u∗, however, random interlacements on Z d, for large d, ought to exhibit similarities to random interlacements on a (2d)regular tree, where the corresponding critical parameter can be explicitly computed, see [19]. We show in this article that lim infd u∗/log d ≥ 1. This lower bound is in agreement with the above mentioned heuristics.
DECOUPLING INEQUALITIES AND INTERLACEMENT PERCOLATION ON G × Z
, 2010
"... We study the percolative properties of random interlacements on G×Z, where G is a weighted graph satisfying certain subGaussian estimates attached to the parameters α> 1 and 2 ≤ β ≤ α + 1. We develop decoupling inequalities, which are a key tool in showing that the critical level u ∗ for the percol ..."
Abstract

Cited by 3 (1 self)
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We study the percolative properties of random interlacements on G×Z, where G is a weighted graph satisfying certain subGaussian estimates attached to the parameters α> 1 and 2 ≤ β ≤ α + 1. 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 setup, and that it is positive when α ≥ 1 + β 2. We also obtain several stretched exponential controls both in the percolative and nonpercolative phases of the model. Even in the case where G = Zd, d ≥ 2, several of these results are new.