Large Deviation Principle for One Dimensional Vector Spin Models with Kac Potentials

Abstract

We consider the one-dimensional planar rotator and classical Heisenberg models with a ferromagnetic Kac potential Jγ(r)=γJ(yr), J with compact support. Below the Lebowitz-Penrose critical temperature the limit (mean-field) theory exhibits a phase transition with a continuum of equilibrium states, indexed by the magnetization vectors mβs, s any unit vector and mβ the Curie–Weiss spontaneous magnetization. We prove a large-deviation principle for the associated Gibbs measures. Then we study the system in the limit γ ↓ 0 below the above critical temperature. We prove that the norm of the empirical spin average in blocks of order γ−1 converges to mβ, uniformly in intervals of order γ−p, for any p ≥ 1. We also give a lower bound to the scale on which the change of phase occurs, by showing that the empirical spin average is approximately constant on intervals having length of order γ-1-λwith λ∈(0,1) small enough.