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].
Categories: math.PR