  On the example of a mean-field Fredrickson-Andersen kinetically constrained
model, we focus on the known property that equilibrium dynamics take place at a
first-order dynamical phase transition point in the space of time-realizations.
We investigate the finite-size properties of this first order transition. By
discussing and exploiting a mapping of the classical dynamical transition -an
argued glassiness signature- to a first-order quantum transition, we show that
the quantum analogy can be exploited to extract finite-size properties, which
in many respects are similar to those in genuine mean-field quantum systems
with a first-order transition. We fully characterize the finite-size properties
of the order parameter across the first order transition.
