# 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 \(2+\alpha \), \(\alpha >0\).

## Keywords

Lattice systems Continuous spin Long-range interaction Decay of correlations Gibbs measure Phase transition Logarithmic Sobolev inequality## Mathematics Subject Classification (2000)

82C26 82B20 60K35 26D10## Notes

### Acknowledgments

Both authors want to thank the anonymous referee for the plentiful and valuable comments on the manuscript. The first author wants to thank Maria Westdickenberg (neé Reznikoff), Felix Otto, Nobuo Yoshida, and Chris Henderson for the fruitful and inspiring discussions on this topic. The second author is grateful to Felix Otto for bringing this subject to his attention. Both authors are indebted to the *Max-Planck Institute for Mathematics in the Sciences* in Leipzig for funding during the years 2010 to 2012, where most of the content of this article originated.

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