Abstract not found

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.

Abstract not found

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∗.

structure induced by a random walk on a

Abstract not found

We consider a random walk on the discrete cylinder (Z/NZ) d ×Z, d ≥ 3 with drift N−dα in the Z-direction and investigate the large N-behavior of the disconnection time T disc N, defined as the first time when the trajectory of the random walk disconnects the cylinder into two infinite components. We prove that, as long as the drift exponent α is strictly greater than 1, the asymptotic behavior of T disc N remains N2d+o(1) , as in the unbiased case considered by Dembo and Sznitman, whereas for α<1, the asymptotic behavior of T disc N becomes exponential in N.

We study the percolative properties of random interlacements on G × Z, where G is a weighted graph satisfying certain sub-Gaussian 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 set-up, and that it is positive when α ≥ 1 + β 2. We also obtain several stretched exponential controls both in the percolative and non-percolative phases of the model. Even in the case where G = Zd, d ≥ 2, several of these results are new.

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