  There are deep analogies between the melting dynamics in systems with a first
order phase transition and the dynamics from equilibrium in super-cooled
liquids. For a class of Ising spin models undergoing a first order transition -
namely p-spin models on the so-called Nishimori line - it can be shown that the
melting dynamics can be exactly mapped to the equilibrium dynamics. In this
mapping the dynamical -or mode-coupling- glass transition corresponds to the
spinodal point, while the Kauzmann transition corresponds to the first order
phase transition itself. Both in mean field and finite dimensional models this
mapping provides an exact realization of the random first order theory scenario
for the glass transition. The corresponding glassy phenomenology can then be
understood in the framework of a standard first order phase transition.
