Local percolative properties of the vacant set of random interlacements with small intensity

TL;DR

This paper investigates the local percolative behavior of the vacant set of random interlacements at low intensity across all dimensions d>=3, providing bounds on the likelihood of large vacant components and introducing a new approach based on conditional independence.

Contribution

It extends previous results to all dimensions d>=3, offering a novel method that leverages conditional independence to analyze local percolation properties at small u.

Findings

Finite vacant components are unlikely to be large at small u.

Provides stretched exponential bounds on the probability of large vacant regions.

Results apply to all dimensions d>=3, not just d>=5.

Abstract

Random interlacements at level u is a one parameter family of connected random subsets of Z^d, d>=3 introduced in arXiv:0704.2560. Its complement, the vacant set at level u, exhibits a non-trivial percolation phase transition in u, as shown in arXiv:0704.2560 and arXiv:0808.3344, and the infinite connected component, when it exists, is almost surely unique, see arXiv:0805.4106. In this paper we study local percolative properties of the vacant set of random interlacements at level u for all dimensions d>=3 and small intensity parameter u>0. We give a stretched exponential bound on the probability that a large (hyper)cube contains two distinct macroscopic components of the vacant set at level u. Our results imply that finite connected components of the vacant set at level u are unlikely to be large. These results were proved in arXiv:1002.4995 for d>=5. Our approach is different from that of arXiv:1002.4995 and works for all d>=3. One of the main ingredients in the proof is a certain conditional independence property of the random interlacements, which is interesting in its own right.