Connectivity Bounds for the Vacant Set of Random Interlacements

Abstract

The model of random interlacements on Z^d, d bigger or equal to 3, was recently introduced in arXiv:0704.2560. A non-negative parameter u parametrizes the density of random interlacements on Z^d. In the present note we investigate the connectivity properties of the vacant set left by random interlacements at level u, in the non-percolative regime, where u is bigger than the non-degenerate critical parameter for percolation of the vacant set, see arXiv:0704.2560, arXiv:0808.3344. We prove a stretched exponential decay of the connectivity function for the vacant set at level u, when u is bigger than an other critical parameter. It is presently an open problem whether these two critical parameters actually coincide.

16 pages, 1 figure, accepted for publication in Ann. Inst. H. Poincare

References in corpus (2)

• On the uniqueness of the infinite cluster of the vacant set of random interlacements
• Upper bound on the disconnection time of discrete cylinders and random interlacements

Cited by in corpus (9)

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• Decoupling inequalities and interlacement percolation on G x Z
• On the fragmentation of a torus by random walk
• Random interlacements and the Gaussian free field
• Percolation in the vacant set of Poisson cylinders
• A lower bound for disconnection by random interlacements
• A short proof of the phase transition for the vacant set of random interlacements
• A lower bound for disconnection by simple random walk
• On the critical parameter of interlacement percolation in high dimension