  We study the most probable way an interface moves on a macroscopic scale from
an initial to a final position within a fixed time in the context of large
deviations for a stochastic microscopic lattice system of Ising spins with Kac
interaction evolving in time according to Glauber (non-conservative) dynamics.
Such interfaces separate two stable phases of a ferromagnetic system and in the
macroscopic scale are represented by sharp transitions. We derive quantitative
estimates for the upper and the lower bound of the cost functional that
penalizes all possible deviations and obtain explicit error terms which are
valid also in the macroscopic scale. Furthermore, using the result of a
companion paper about the minimizers of this cost functional for the
macroscopic motion of the interface in a fixed time, we prove that the
probability of such events can concentrate on nucleations should the transition
happen fast enough.
