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

TL;DR

This paper investigates properties of the vacant set in random interlacements for dimensions at least 5 and small intensity, providing bounds on the size of finite clusters and demonstrating the ubiquity of the infinite component.

Contribution

It establishes stretched exponential bounds on separation probabilities and analyzes the distribution of finite cluster sizes in the vacant set at small intensities.

Findings

Bound on the probability of interlacement set separating large sets

Distribution estimates for the diameter and volume of finite vacant clusters

High probability of the infinite component being ubiquitous near the origin

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.