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 (Commun Math Phys 347(3):903–931, 2016) and Aza–Bru–Siqueira Pedra (Commun Math Phys 360(2):715–726, 2018). 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 (Commun Math Phys 316(3):641–692, 2012) .
