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Vacant set of random interlacements and percolation
"... We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z d, d ≥ 3. A nonnegative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when conside ..."
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Cited by 72 (15 self)
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We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z d, d ≥ 3. A nonnegative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder (Z/NZ) d−1 × Z by simple random walk, or the set of points visited by simple random walk on the discrete torus (Z/NZ) d at times of order uN d. In particular we study the percolative properties of the vacant set left by the interlacement at level u, which is an infinite connected translation invariant random subset of Z d. We introduce a critical value u ∗ such that the vacant set percolates for u < u ∗ and does not percolate for u> u∗. Our main results show that u ∗ is finite when d ≥ 3 and strictly positive when d ≥ 7.
RANDOM WALK ON A DISCRETE TORUS AND RANDOM INTERLACEMENTS
 ELECTRONIC COMMUNICATIONS IN PROBABILITY
, 2008
"... We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ) d, d ≥ 3, until uN d time steps, u> 0, and the model of random interlacements recently introduced by Sznitman [9]. In particular, we show that for large N, the joint distributi ..."
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Cited by 36 (4 self)
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We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ) d, d ≥ 3, until uN d time steps, u> 0, and the model of random interlacements recently introduced by Sznitman [9]. In particular, we show that for large N, the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to time uN d converges to independent copies of the random interlacement at level u.
A lower bound on the disconnection time of a discrete cylinder
 In and Out of Equilibrium 2, Birkh"auser
, 2008
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On the uniqueness of the infinite cluster of the vacant set of random interlacements
 ANN. APPL. PROBAB
, 2009
"... We consider the model of random interlacements on Z d introduced in Sznitman [Vacant set of random interlacements and percolation (2007) preprint]. For this model, we prove the uniqueness of the infinite component of the vacant set. As a consequence, we derive the continuity in u of the probability ..."
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Cited by 25 (4 self)
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We consider the model of random interlacements on Z d introduced in Sznitman [Vacant set of random interlacements and percolation (2007) preprint]. For this model, we prove the uniqueness of the infinite component of the vacant set. As a consequence, we derive the continuity in u of the probability that the origin belongs to the infinite component of the vacant set at level u in the supercritical phase u<u∗.
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 25 (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.
QUENCHED SCALING LIMITS OF TRAP MODELS
"... Abstract. Fix a strictly positive measure W on the ddimensional torus Td. For an integer N ≥ 1, denote by W N x, x = (x1,..., xd), 0 ≤ xi < N, the Wmeasure of the cube [x/N, (x+1)/N), where 1 is the vector with all components equal to 1. In dimension 1, we prove that the hydrodynamic behavior o ..."
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Cited by 13 (10 self)
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Abstract. Fix a strictly positive measure W on the ddimensional torus Td. For an integer N ≥ 1, denote by W N x, x = (x1,..., xd), 0 ≤ xi < N, the Wmeasure of the cube [x/N, (x+1)/N), where 1 is the vector with all components equal to 1. In dimension 1, we prove that the hydrodynamic behavior of a superposition of independent random walks, in which a particle jumps from x/N to one of its neighbors at rate (NW N x) −1, is described in the diffusive scaling by the linear differential equation ∂tρ = (d/dW)(d/dx)ρ. In dimension d> 1, if W is a finite discrete measure, W = P i≥1 wiδxi, we prove that the random walk which jumps from x/N uniformly to one of its neighbors at rate (W N x) −1 has a metastable behavior, as defined in [2], described by the Kprocess introduced in [13]. 1.
RANDOM PLANAR METRICS
"... Abstract. A discussion regarding aspects of several quite different random planar metrics and related topics is presented. 1. ..."
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Cited by 12 (3 self)
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Abstract. A discussion regarding aspects of several quite different random planar metrics and related topics is presented. 1.
Deterministic random walks
 In Proceedings of the Workshop on Analytic Algorithmics and Combinatorics
, 2006
"... structure of the vacant set induced by a ..."
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