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On the domination of random walk on a discrete cylinder by random interlacements
 Electron. J. Probab
"... We consider simple random walk on a discrete cylinder with base a large ddimensional torus of sidelength N, when d ≥ 2. We develop a stochastic domination control on the local picture left by the random walk in boxes of sidelength of order N 1−ε, with 0 < ε < 1, at certain random times comp ..."
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Cited by 7 (3 self)
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We consider simple random walk on a discrete cylinder with base a large ddimensional torus of sidelength N, when d ≥ 2. We develop a stochastic domination control on the local picture left by the random walk in boxes of sidelength of order N 1−ε, with 0 < ε < 1, at certain random times comparable to N 2d, in terms of the trace left in a similar box of Z d+1 by random interlacements at a suitably adjusted level. As an application we derive a lower bound on the disconnection time TN of the discrete cylinder, which as a byproduct shows the tightness of the laws of N 2d /TN, for all d ≥ 2. This fact had previously only been established when d ≥ 17, in [3].
Connectivity bounds for the vacant set of random interlacements, preprint available at http://www.math.ethz.ch/u/sznitman/preprints
"... The model of random interlacements on Z d, d ≥ 3, was recently introduced in [4]. A nonnegative parameter u parametrizes the density of random interlacements on Z d. In the present note we investigate connectivity properties of the vacant set left by random interlacements at level u, in the nonper ..."
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Cited by 6 (4 self)
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The model of random interlacements on Z d, d ≥ 3, was recently introduced in [4]. A nonnegative parameter u parametrizes the density of random interlacements on Z d. In the present note we investigate connectivity properties of the vacant set left by random interlacements at level u, in the nonpercolative regime u> u∗, with u ∗ the nondegenerate critical parameter for the percolation of the vacant set, see [4], [3]. We prove a stretched exponential decay of the connectivity function for the vacant set at level u, when u> u∗∗, where u∗ ∗ is another critical parameter introduced in [6]. It is presently an open problem whether u∗ ∗ actually coincides with u∗.
The effect of small quenched noise on connectivity properties of random interlacements
, 2013
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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 per ..."
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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.
PHASE TRANSITION AND LEVELSET PERCOLATION FOR THE GAUSSIAN FREE FIELD
, 2012
"... We consider levelset percolation for the Gaussian free field on Z d, d ≥ 3, and prove that, as h varies, there is a nontrivial percolation phase transition of the excursion set above level h for all dimensions d ≥ 3. So far, it was known that the corresponding critical level h∗(d) satisfies h∗(d) ..."
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Cited by 1 (1 self)
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We consider levelset percolation for the Gaussian free field on Z d, d ≥ 3, and prove that, as h varies, there is a nontrivial percolation phase transition of the excursion set above level h for all dimensions d ≥ 3. So far, it was known that the corresponding critical level h∗(d) satisfies h∗(d) ≥ 0 for all d ≥ 3 and that h∗(3) is finite, see [2]. We prove here that h∗(d) is finite for all d ≥ 3. In fact, we introduce a second critical parameter h∗ ∗ ≥ h∗, show that h∗∗(d) is finite for all d ≥ 3, and that the connectivity function of the excursion set above level h has stretched exponential decay for all h> h∗∗. Finally, we prove that h ∗ is strictly positive in high dimension. It remains open whether h ∗ and h∗ ∗ actually coincide and whether h ∗> 0 for all d ≥ 3.