No phase transition for Gaussian fields with bounded spins

Let a < b, = [a, b]Z and H be the (formal) Hamiltonian defined on by H(η)= 1 2 ∑ x,y∈Zd J (x − y)(η(x)− η(y)), (1) where J : Z → R is any summable non-negative symmetric function (J (x) ≥ 0 for all x ∈ Z , x J (x) <∞ and J (x)= J (−x)). We prove that there is a unique Gibbs measure on associated to H . The result is a consequence of the fact that the corresponding Gibbs sampler is attractive and has a unique invariant measure.