Stephen Bigelis, Emilio N.M. Cirillo, Joel L. Lebowitz, Eugene R. Speer
Critical droplets in Metastable States of Probabilistic Cellular Automata
(53K, LATeX)
ABSTRACT. We consider the problem of metastability in a
probabilistic cellular automaton (PCA) with a parallel updating rule which
is reversible with respect to a Gibbs measure. The dynamical rules
contain two parameters $\beta$ and $h$ which resemble, but are not
identical to, the inverse temperature and external magnetic field in a
ferromagnetic Ising model; in particular, the phase diagram of the system
has two stable phases when $\beta$ is large enough and $h$ is zero, and a
unique phase when $h$ is nonzero. When the system evolves, at small
positive values of $h$, from an initial state with all spins down, the PCA
dynamics give rise to a transition from a metastable to a stable phase when
a droplet of the favored $+$ phase inside the metastable $-$ phase reaches
a critical size. We give heuristic arguments to estimate the critical size
in the limit of zero ``temperature'' ($\beta\to\infty$), as well as
estimates of the time required for the formation of such a droplet in a
finite system. Monte Carlo simulations give results in good agreement with
the theoretical predictions.