No Phase Transition for Gaussian Fields with Bounded Spins

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Let a<b, $\Omega=[a,b]^{{\mathbb{Z}}^{d}}$ and H be the (formal) Hamiltonian defined on Ω by 1 $$\label{a1}H(\eta)=\frac{1}{2}\sum_{x,y\in {\mathbb{Z}}^{d}}J(x-y)(\eta(x)-\eta(y))^{2},$$ 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.