Exponential decay of truncated correlations for the Ising model in any dimension for all but the critical temperature

TL;DR

This paper proves that the truncated two-point correlations in the ferromagnetic Ising model decay exponentially fast in all non-critical phases across any dimension, confirming exponential clustering except at the critical point.

Contribution

It establishes exponential decay of correlations in the Ising model for all dimensions greater than or equal to three, outside the critical temperature, extending previous results.

Findings

Exponential decay of correlations in the ordered phase for all dimensions $d \\ge 3$.

Exponential clustering holds throughout the phase diagram except at the critical point.

The result applies to the pure phases with zero external magnetic field.

Abstract

The truncated two-point function of the ferromagnetic Ising model on $\mathbb Z^d$ ($d\ge3$) in its pure phases is proven to decay exponentially fast throughout the ordered regime ($\beta>\beta_c$ and $h=0$). Together with the previously known results, this implies that the exponential clustering property holds throughout the model's phase diagram except for the critical point: $(\beta,h) = (\beta_c,0)$.