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70
A lower bound on the disconnection time of a discrete cylinder
 In and Out of Equilibrium 2, Birkh"auser
, 2008
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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 23 (4 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.
GIANT VACANT COMPONENT LEFT BY A RANDOM WALK IN A RANDOM dREGULAR GRAPH
, 2009
"... We study the trajectory of a simple random walk on a dregular graph with d ≥ 3 and locally treelike structure as the number n of vertices grows. Examples of such graphs include random dregular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of ..."
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Cited by 21 (4 self)
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We study the trajectory of a simple random walk on a dregular graph with d ≥ 3 and locally treelike structure as the number n of vertices grows. Examples of such graphs include random dregular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u> 0 is a fixed positive parameter. We show that this socalled vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u ⋆ ∈ (0, ∞) such that, with high probability as n grows, if u < u⋆, then the largest component of the vacant set has a volume of order n, and if u> u⋆, then it has a volume of order log n. The critical value u ⋆ coincides with the critical intensity of a random interlacement process on a dregular tree. We also show that the random interlacements model describes the structure of the vacant set in local neighbourhoods.
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 20 (6 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].
An isomorphism theorem for random interlacements
 Electron. Commun. Probab
, 2012
"... We consider continuoustime random interlacements on a transient weighted graph. We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian free field on the weighted graph. This identity is closely linked to the generalized second RayKnig ..."
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Cited by 17 (5 self)
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We consider continuoustime random interlacements on a transient weighted graph. We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian free field on the weighted graph. This identity is closely linked to the generalized second RayKnight theorem of [2], [4], and uniquely determines the law of occupation times of random interlacements at level u.
Connectivity bounds for the vacant set of random interlacements
, 2009
"... 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 16 (7 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∗.
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 15 (2 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.
The effect of small quenched noise on connectivity properties of random interlacements
, 2013
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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 ..."
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Cited by 11 (1 self)
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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.