Strong local uniqueness for the vacant set of random interlacements

We consider the the vacant set ${\mathcal{V}}^u$ of random interlacements on ${\mathbb{Z}}^d$ in dimensions $d\ge 3$. For varying intensity $u>0$, the connectivity properties of ${\mathcal{V}}^u$ undergo a percolation phase transition across a critical parameter $u_{\ast }\in (0,\mathrm{\infty })$. In this article, we prove that this phase transition is sharp in the supercritical phase $u<u_{\ast }$. This follows from a certain strong local uniqueness property (SLU) introduced in the present work, which we prove ${\mathcal{V}}^u$ satisfies. In itself, this property furnishes the missing ingredient needed to deduce a number of desirable quenched results characterizing the large-scale geometry of the infinite cluster. Moreover, SLU entails a sought-after local and monotone criterion amenable to renormalization arguments below $u_{\ast }$.