Disconnection, random walks, and random interlacements

Abstract

We consider random interlacements on Z^d, with d bigger or equal to 3, when their vacant set is in a strongly percolative regime. We derive an asymptotic upper bound on the probability that the random interlacements disconnect a box of large side-length from the boundary of a larger homothetic box. As a corollary, we obtain an asymptotic upper bound on a similar quantity, where the random interlacements are replaced by the simple random walk. It is plausible, but open at the moment, that these asymptotic upper bounds match the asymptotic lower bounds obtained by Xinyi Li and the author in arXiv:1310.2177, for random interlacements, and by Xinyi Li in a recent article, for the simple random walk. In any case, our bounds capture the principal exponential rate of decay of these probabilities, in any dimension d bigger or equal to 3.

38 pages, corresponds to the version published in Probability Theory and Related Fields except for the numbering of the sections

References in corpus (2)

• A lower bound for disconnection by random interlacements
• On the disconnection of a discrete cylinder by a biased random walk

Cited by in corpus (12)

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• A lower bound for disconnection by simple random walk
• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• On the cost of the bubble set for random interlacements
• Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
• Phase transition for level-set percolation of the membrane model in dimensions $d\ge 5$
• Random interlacement is a factor of i.i.d
• First passage percolation, local uniqueness for interlacements and capacity of random walk
• Entropic repulsion for the occupation-time field of random interlacements conditioned on disconnection