Decay of Correlations in 1D Lattice Systems of Continuous Spins and Long-Range Interaction

Abstract

We consider a one-dimensional lattice system of unbounded and continuous spins. The Hamiltonian consists of a perturbed strictly-convex single-site potential and a product term with longe-range interaction. We show that if the interactions have an algebraic decay of order \(2+\alpha \), \(\alpha >0\), then the correlations also decay algebraically of order \(2+ \tilde{\alpha }\) for some \(\alpha > \tilde{\alpha }> 0\). For the argument we generalize a method due to Zegarlinski from finite-range to infinite-range interaction to get a preliminary decay of correlations, which is improved to the correct order by a recursive scheme based on Lebowitz inequalities. Because the decay of correlations yields the uniqueness of the Gibbs measure, the main result of this article yields that the one-phase region of a continuous spin system is at least as large as for the Ising model. This shows that there is no-phase transition in one-dimensional systems of unbounded and continuous spins as long as the interaction decays algebraically of order \(2+\alpha \), \(\alpha >0\).