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, ${\textsf {F}}$ and its translated ${\textsf {F}}+x$, 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 ${\textsf {F}}$ and ${\textsf {F}}+x$, respectively. This improves a previous bound obtained by Sznitman (Ann Math 2(171):2039–2087, 2010).