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arXiv · 1002.4995

On the size of a finite vacant cluster of random interlacements with small intensity

Augusto Teixeira
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Abstract

In this paper we establish some properties of percolation for the vacant set of random interlacements, for d at least 5 and small intensity u. The model of random interlacements was first introduced by A.S. Sznitman in arXiv:0704.2560. It is known that, for small u, almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u, see arXiv:0808.3344 and arXiv:0805.4106. We estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin, given that it is finite. This comes as a by-product of our main theorem, which proves a stretched exponential bound on the probability that the interlacement set separates two macroscopic connected sets in a large cube. As another application, we show that with high probability, the unique infinite connected component of the vacant set is `ubiquitous' in large neighborhoods of the origin.

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Augusto Teixeira. 2010-02-26. On the size of a finite vacant cluster of random interlacements with small intensity. https://arxiv.org/abs/1002.4995

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