Large-Deviation Principle for One-Dimensional Vector Spin Models with Kac Potentials |
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Authors: | Paolo Buttà Pierre Picco |
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Affiliation: | (1) INRIA CMI, Université de Provence, 39, F-13453 Marseille Cedex 13, France;(2) Present address: Center for Mathematical Sciences Research, Rutgers, the, State University of New Jersey, Piscataway, New jersey, 08854-8019;(3) CPT-CNRS, Luminy, Case 907, F-13288 Marseille Cedex 9, France |
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Abstract: | We consider the one-dimensional planar rotator and classical Heisenberg models with a ferromagnetic Kac potential J(r)=J(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 ms, s any unit vector and m the Curie–Weiss spontaneous magnetization. We prove a large-deviation principle for the associated Gibbs measures. Then we study the system in the limit 0 below the above critical temperature. We prove that the norm of the empirical spin average in blocks of order –1 converges to m, uniformly in intervals of order –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 -1-with (0,1) small enough. |
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Keywords: | Large deviations Kac potentials spin vector models |
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