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

TL;DR

This paper proves that in a one-dimensional lattice system with continuous, unbounded spins and algebraically decaying long-range interactions, correlations decay at a similar algebraic rate, ensuring uniqueness of the Gibbs measure.

Contribution

It extends correlation decay results from finite-range to infinite-range interactions using a recursive scheme, establishing the size of the on-phase region for continuous spins.

Findings

Correlations decay algebraically with the same order as interactions

Uniqueness of the Gibbs measure in the on-phase region

Method generalizes Zegarlinski's approach to infinite-range interactions

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+\~a for some \~a > 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.