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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 26 (11 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.
A lower bound on the disconnection time of a discrete cylinder
 In and Out of Equilibrium 2, Birkh"auser
, 2008
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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 18 (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.
Percolation for the vacant set of random interlacements
 Comm. Pure Appl. Math
"... Vladas Sidoravicius would like to thank the FIM for financial support and hospitality during his visitsto ETH. His research was also partially supported by CNPq and FAPERJ. ..."
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Cited by 15 (9 self)
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Vladas Sidoravicius would like to thank the FIM for financial support and hospitality during his visitsto ETH. His research was also partially supported by CNPq and FAPERJ.
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 12 (2 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∗.
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 9 (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 7 (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∗.