An improved decoupling inequality for random interlacements

In this paper we obtain a decoupling feature of the random interlacements process $mathcal{I}^u subset mathbb{Z}^d$, at level $u$, $dgeq 3$. 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].