  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, $\beta^{-1}$, and magnitude,
$\theta$, of the field in the region where the free energy of the corresponding
random Curie Weiss model has only two absolute minima $m_\beta$ and $Tm_\beta$.
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 ${\cal F}^*$ to undergo a phase change (the surface
tension). The ${\cal F}^*$ 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 $m_\beta$ and $Tm_\beta$. 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].
