Weak large deviations principle for one dimensional random field Kac model

We present 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 Bernouilli random variables. The results are valid for values of the temperature and magnitude of the field in the region where the free energy of the corresponding random Curie Weiss model has only two absolute minimizers. We give an explicit representation of the large deviation rate function and characterize its minimizers. We show that they are step functions taking two values, the two absolute minimizers of the free energy of the random Curie Weiss model. The points of discontinuity are described by a stationary renewal process related to the $h-$extrema of a bilateral Brownian motion studied by Neveu and Pitman, where $h$ depends on the temperature and magnitude of the random field. Our result is a complete characterization of the typical profiles of RFKM (the ground states) which was initiated in [\rcite{COP1}] and extended in [\rcite{COPV}]