  We revisit the phenomenon of spinodals in the presence of quenched disorder
and develop a complete theory for it. We focus on the spinodal of an Ising
model in a quenched random field (RFIM), which has applications in many areas
from materials to social science. By working at zero temperature in the
quasi-statically driven RFIM, thermal fluctuations are eliminated and one can
give a rigorous content to the notion of spinodal. We show that the latter is
due to the depinning and the subsequent expansion of rare droplets. We work out
the associated critical behavior, which, in any finite dimension, is very
different from the mean-field one: the characteristic length diverges
exponentially and the thermodynamic quantities display very mild
nonanalyticities much like in a Griffith phenomenon. From the recently
established connection between the spinodal of the RFIM and glassy dynamics,
our results also allow us to conclusively assess the physical content and the
status of the dynamical transition predicted by the mean-field theory of
glass-forming liquids.
