Connectivity Bounds for the Vacant Set of Random Interlacements

TL;DR

This paper studies the connectivity properties of the vacant set left by random interlacements on high-dimensional integer lattices, proving a stretched exponential decay of the connectivity function in the non-percolative regime.

Contribution

It establishes a stretched exponential decay of the vacant set connectivity function for levels above a certain critical parameter, advancing understanding of the non-percolative phase.

Findings

Proves stretched exponential decay of connectivity function

Identifies a critical parameter for non-percolative regime

Highlights open problem on the equality of critical parameters

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.