Enza Orlandi and Pierre Picco
One-dimensional random field Kac's model:
weak large deviations principle
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ABSTRACT. Abstract We prove a quenched weak large deviations principle for the Gibbs measures of a Random Field Kac Model (RFKM) in one dimension. The external random magnetic field is given by symmetrically distributed Bernoulli random variables. The results are valid for values of the temperature, ^{ 1}, and magnitude, , of the field in the region where the free energy of the corresponding random Curie Weiss model has only two absolute minima m_ and Tm_ . We give an explicit representation of the rate functional which is a positive random functional determined by two distinct contributions. One is related to the free energy cost F to undergo a phase change (the surface tension). The F is the cost of one single phase change and depends on the temperature and magnitude of the field. The other is a bulk contribution due to the presence of the random magnetic field. We characterize the minimizers of this random functional. We show that they are step functions taking values m_ and Tm_ . The points of discontinuity are described by a stationary renewal process related to the h extrema for a bilateral Brownian motion studied by Neveu and Pitman, where h in our context is a suitable constant depending on the temperature and on magnitude of the random field. As an outcome we have a complete characterization of the typical profiles of RFKM (the ground states) which was initiated in [14] and extended in [16].