Weighted dependency graphs and the Ising model

TL;DR

This paper demonstrates that spins in the d-dimensional Ising model exhibit a weighted dependency structure, enabling the derivation of central limit theorems for pattern counts in large regions.

Contribution

It establishes that the Ising model's spins have a weighted dependency graph, facilitating new central limit theorems for pattern occurrences.

Findings

Weighted dependency graphs apply to the Ising model.

Central limit theorems are proved for pattern counts.

Results hold for growing regions in the model.

Abstract

Weighted dependency graphs have been recently introduced by the second author, as a toolbox to prove central limit theorems. In this paper, we prove that spins in the $d$-dimensional Ising model display such a weighted dependency structure. We use this to obtain various central limit theorems for the number of occurrences of local and global patterns in a growing box.