# Decay of Multi-point Correlation Functions in ℤd

## TL;DR

This paper proves new bounds on multi-point correlation functions in mathematical lattices and applies them to models in statistical physics.

## Contribution

The paper provides the first examples of multi-point dynamical localization in disordered systems as conjectured by Bravyi–König.

## Key findings

- Multi-point correlation bounds are established for arbitrary dimensions in ℤd with symmetrized distances.
- The results are applied to the Ising model on ℤd and to prove multi-point dynamical localization in disordered systems.
- These findings resolve open questions from previous mathematical physics literature.

## Abstract

We prove multi-point correlation bounds in \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}^d$$\end{document}Zd for arbitrary \documentclass[12pt]{minimal}
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\begin{document}$$d\ge 1$$\end{document}d≥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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}^d$$\end{document}Zd, 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) .