Large-deviation principle for one-dimensional vector spin models with Kac potentials
This paper studies planar rotor and classical Heisenberg models with Kac interactions in one dimension. The results concern first the classical Lebowitz-Penrose theorem, that states that the free energy as a function of the total magnetization vector converges to the convex hull of the corresponding function of the related mean field model. The second, and more difficult result of the paper is an upper estimate on the scale over which the ``mesoscopic magnetization profiles'' are Gibbs-typically constant. If \(\gamma^{-1}\) is the range of the interaction, it is found that this scale is larger than \(\gamma^{-1-\lambda}\) with some positive \(\lambda\). (Note that it is expected that such a result should be true with \(\lambda=1\).) Results of this type have first been obtained in the Kac-Ising model by \textit{M. Cassandro, E. Orlandi} and \textit{E. Presutti} [Probab. Theory Relat. Fields 96, No. 1, 57-96 (1993; Zbl 0791.60096)].
scientific article; zbMATH DE number 1357842