Weighted dependency graphs and the Ising model

Jehanne Dousse, Valentin Féray

Published 2016-10-17Version 1

We consider the Ising model on the $d$-dimensional square lattice at either very high temperature, very low temperature or in a strong magnetic field. In each of these three regimes, we give a simple proof that the joint cumulants of spins decay exponentially fast with respect to the tree-length of the set of spins. With the recent terminology of weighted dependency graphs introduced by the second author, this means that we have a weighted dependency graph structure on the spins. We use this structure to reprove Newman's central limit theorem for the magnetization in a growing box, and to extend it to the number of occurrences of a given spin pattern.