One-dimensional random field Kac's model: weak large deviations principle
Pierre Picco, CNRS
Enza Orlandi, Universita di Roma TRE
Abstract
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
[2] and extended in [4].
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