An improved decoupling inequality for random interlacements

Abstract

In this paper we obtain a decoupling feature of the random interlacements process ${\mathcal{I}}^u\subset {\mathbb{Z}}^d$, at level $u$, $d\ge 3$. More precisely, we show that the trace of the random interlacements process on two disjoint finite sets, $\text{F}$ and its translated $\text{F}+x$, can be coupled with high probability of success, when $\Vert x\Vert$ 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 $\text{F}$ and $\text{F}+x$, respectively. This improves a previous bound obtained by Sznitman in [12].

30 pages, 2 figures, revised and corrected version, added references, accepted for publication in the Journal of Statistical Physics