Decay of correlations in 1D lattice systems of continuous spins and long-range interaction

Abstract

We consider an one-dimensional lattice system of unbounded and continuous spins. The Hamiltonian consists of a perturbed strictly-convex single-site potential and with longe-range interaction. We show that if the interactions decay algebraically of order 2+a, a>0 then the correlations also decay algebraically of order 2+ã for some ã > 0. For the argument we generalize a method from 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 on-phase region of a continuous spin system is at least as large as for the Ising model.

References in corpus (3)

• Phase Transition in the 1d Random Field ising model with long range interaction
• Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential
• The approach of Otto-Reznikoff revisited

Cited by in corpus (6)

• Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature
• Decay of correlations and uniqueness of the infinite-volume Gibbs measure of the canonical ensemble of 1d-lattice systems
• Strict convexity of the free energy of the canonical ensemble under decay of correlations
• The approach of Otto-Reznikoff revisited
• Equivalence of the grand canonical ensemble and the canonical ensemble on 1d-lattice systems
• Hydrodynamic limit of the Kawasaki dynamics on the 1D-lattice with strong, finite-range interaction