The truncated correlations of the Ising model in any dimension decay exponentially fast at all but the critical temperature

TL;DR

This paper proves that in the ferromagnetic Ising model on high-dimensional lattices, the two-point correlations decay exponentially fast in all phases except at the critical temperature, confirming exponential clustering away from criticality.

Contribution

It establishes exponential decay of correlations in the ordered phase for the Ising model in any dimension greater than or equal to three, extending known results to the entire phase diagram except at criticality.

Findings

Exponential decay of two-point functions in the ordered phase.

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

Results apply to the ferromagnetic Ising model in any dimension $d \\ge 3$.

Abstract

The truncated two-point function of the nearest-neighbor ferromagnetic Ising model on $\mathbb Z^d$ ($d\ge3$) in its pure phases is proven to decays exponentially fast throughout the ordered regime ($T<T_c$). Together with known results, this implies that the exponential clustering property holds throughout the model's phase diagram except for the critical point: $(T,h) = (T_c,0)$.