Decay of multi-point correlation functions in ${\mathbb{Z}}^d$

We prove multi-point correlation bounds in ${\mathbb{Z}}^d$ for arbitrary $d\ge 1$ with symmetrized distances, answering open questions proposed by Sims-Warzel \cite{SW} and Aza-Bru-Siqueira Pedra \cite{ABP}. As applications, we prove multi-point correlation bounds for the Ising model on ${\mathbb{Z}}^d$, and multi-point dynamical localization in expectation for uniformly localized disordered systems, which provides the first examples of this conjectured phenomenon by Bravyi-König \cite{BK}.