An improved decoupling inequality for random interlacements

TL;DR

This paper establishes a decoupling inequality for random interlacements in high dimensions, showing that the process on distant sets can be approximated by independent excursions, leading to improved covariance bounds.

Contribution

It introduces a new decoupling inequality for random interlacements, enhancing previous bounds and enabling better understanding of spatial independence in the process.

Findings

Coupling of interlacement traces with independent excursions for large displacements.

Improved upper bounds on covariance between functions of disjoint interlacement configurations.

Enhanced decoupling techniques for analyzing random interlacements in high dimensions.

Abstract

In this paper we obtain a decoupling feature of the random interlacements process $\mathcal{I}^u \subset \mathbb{Z}^d$, at level $u$, $d\geq 3$. More precisely, we show that the trace of the random interlacements process on two disjoint finite sets, $\textsf{F}$ and its translated $\textsf{F}+x$, can be coupled with high probability of success, when $\|x\|$ is large, with the trace of a process of independent excursions, which we call the noodle soup process. As a consequence, we obtain an upper bound on the covariance between two $[0,1]$-valued functions depending on the configuration of the random interlacements on $\textsf{F}$ and $\textsf{F}+x$, respectively. This improves a previous bound obtained by Sznitman in [12].