Exponential Decay of Truncated Correlations for the Ising Model in any Dimension for all but the Critical Temperature

Abstract

The truncated two-point function of the ferromagnetic Ising model on Z d (d ≥ 3) in its pure phases is proven to decay exponentially fast throughout the ordered regime (β > β 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: (β, h) = (β c , 0).

Weak Mixing and Analyticity of the Pressure in the Ising Model

“…• [17] where smoothness of β → ψ(β, 0) is proved in the regime β ≥ β + > β c , the proof works under the assumption that the covariances decay exponentially with the distance in a pure state. Toghether with [6], this implies smoothness in the regime β > β c . It is also shown, under the same hypotheses, that ψ possesses directional derivatives at all orders in h at h = 0.…”

Sharp Asymptotics for the Truncated Two-Point Function of the Ising Model with a Positive Field

“…The first problem suggested by the results of this paper, of [6] and of [14], is the sharp treatment of truncated correlations when h = 0, β > β c , d ≥ 3, as it is the only regime missing. Such an analysis seems doable via the arguments presented in [10] combined with the general approach of [7] and some ideas borrowed from the present work. We plan to come back to this question in a near future.…”

“…Exponential Ratio Mixing when h > 0. We describe here an adaptation of an argument due to Duminil-Copin [9] to obtain exponential mixing under the random current measure with a field (a version of this idea is used in [10]).…”

Asymptotics of even–even correlations in the Ising model

“…Of course, the existence of τ β,h does not imply that the truncated 2-point function actually decays exponentially fast, since τ β,h might in fact equal 0. It turns out, however, that the latter happens only at the critical point: This was shown, for the Ising model with finite-range interactions, in [25] when h = 0, in [3] when h = 0 and β < β c (d), and in [17] when h = 0 and β > β c (d).…”