Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup

TL;DR

This paper establishes weaker decoupling inequalities for the vacant set of the random walk loop soup on or dimensions d  and demonstrates that large-scale geometric properties known for other percolation models also hold for this model, including the non-triviality of the supercritical regime.

Contribution

It introduces weaker decoupling inequalities for the random walk loop soup's vacant set and extends known geometric results to this model, which was previously not covered.

Findings

Weaker decoupling inequalities hold for the vacant set of the random walk loop soup.

Large-scale geometric properties from other percolation models apply to this model.

The strongly supercritical regime for the vacant set is proven to be non-trivial.

Abstract

It has been recently understood (arXiv:1212.2885, arXiv:1310.4764, arXiv:1410.0605) that for a general class of percolation models on $\mathbb{Z}^d$ satisfying suitable decoupling inequalities, which includes i.a.\ Bernoulli percolation, random interlacements and level sets of the Gaussian free field, large scale geometry of the unique infinite cluster in strongly percolative regime is qualitatively the same; in particular, the random walk on the infinite cluster satisfies the quenched invariance principle, Gaussian heat-kernel bounds and local CLT. In this paper we consider the random walk loop soup on $\mathbb{Z}^d$ in dimensions $d\geq 3$. An interesting aspect of this model is that despite its similarity and connections to random interlacements and the Gaussian free field, it does not fall into the above mentioned general class of percolation models, since the required decoupling inequalities are not valid. We identify weaker (and more natural) decoupling inequalities and prove that (a) they do hold for the random walk loop soup and (b) all the results about the large scale geometry of the infinite percolation cluster proved for the above mentioned class of models hold also for models that satisfy the weaker decoupling inequalities. Particularly, all these results are new for the vacant set of the random walk loop soup. (The range of the random walk loop soup has been addressed by Chang arXiv:1504.07906 by a model specific approximation method, which does not apply to the vacant set.) Finally, we prove that the strongly supercritical regime for the vacant set of the random walk loop soup is non-trivial. It is expected, but open at the moment, that the strongly supercritical regime coincides with the whole supercritical regime.