No Phase Transition for Gaussian Fields with Bounded Spins

Abstract

Let a<b, $\Omega=[a,b]^{{\mathbb{Z}}^{d}}$ and H be the (formal) Hamiltonian defined on Ω by

$$\label{a1}H(\eta)=\frac{1}{2}\sum_{x,y\in {\mathbb{Z}}^{d}}J(x-y)(\eta(x)-\eta(y))^{2},$$

(1)

where J:ℤ^d→ℝ is any summable non-negative symmetric function (J(x)≥0 for all x∈ℤ^d, ∑ _x J(x)<∞ and J(x)=J(−x)). We prove that there is a unique Gibbs measure on Ω associated to H. The result is a consequence of the fact that the corresponding Gibbs sampler is attractive and has a unique invariant measure.