Critical droplets in Metastable States of Probabilistic Cellular Automata

TL;DR

This paper investigates the metastable behavior of a probabilistic cellular automaton with parameters analogous to temperature and magnetic field, analyzing critical droplet formation and transition times through heuristic estimates and simulations.

Contribution

It provides new heuristic estimates for critical droplet size and formation time in a PCA model, supported by Monte Carlo simulations.

Findings

Critical droplet size increases as temperature decreases.

Formation time of droplets scales with system parameters.

Simulation results align with theoretical predictions.

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.