Cylinders' percolation: decoupling and applications

TL;DR

This paper establishes a strong decoupling inequality for the complex cylinder percolation process, enabling analysis of its dependency structure and demonstrating that, at low densities, the vacant set's random walk is transient in dimensions three and higher.

Contribution

It introduces a new notion of fast decoupling for the cylinder percolation model and proves its validity, facilitating the study of dependencies and applications like transience of random walks.

Findings

Decoupling inequality proven for cylinder percolation.

At low density, the vacant set's random walk is transient in dimensions ≥ 3.

The model features infinite-range, rigid cylinders with strong dependencies.

Abstract

In this paper we establish a strong decoupling inequality for the cylinder's percolation process introduced by Tykesson and Windisch in arXiv:1010.5338 . This model features a very strong dependency structure, making it difficult to study, and this is why such decoupling inequalities are desirable. It is important to notice that the type of dependencies featured by cylinder's percolation is particularly intricate, given that the cylinders have infinite range (unlike some models like Boolean percolation) while at the same time being rigid bodies (unlike processes such as Random Interlacements). Our work introduces a new notion of fast decoupling, proves that it holds for the model in question and finishes with an application. More precisely, we prove that for a small enough density of cylinders, a random walk on a connected component of the vacant set is transient for all dimensions $d \geq 3$.