Metastability for reversible probabilistic cellular automata with self--interaction

TL;DR

This paper analyzes the metastable behavior of reversible probabilistic cellular automata with self-interaction under small temperature and magnetic field, focusing on exit times and critical droplets in a parallel updating scheme.

Contribution

It introduces a novel approach to characterize metastability in parallel dynamics without fully describing fixed points, using energy barriers and recurrence properties.

Findings

Computed the logarithmic exit time from metastable states.

Identified the critical droplet necessary for phase transition.

Provided a framework for metastability analysis in reversible automata.

Abstract

The problem of metastability for a stochastic dynamics with a parallel updating rule is addressed in the Freidlin--Wentzel regime, namely, finite volume, small magnetic field, and small temperature. The model is characterized by the existence of many fixed points and cyclic pairs of the zero temperature dynamics, in which the system can be trapped in its way to the stable phase. %The characterization of the metastable behavior %of a system in the context of parallel dynamics is a very difficult task, %since all the jumps in the configuration space are allowed. Our strategy is based on recent powerful approaches, not needing a complete description of the fixed points of the dynamics, but relying on few model dependent results. We compute the exit time, in the sense of logarithmic equivalence, and characterize the critical droplet that is necessarily visited by the system during its excursion from the metastable to the stable state. We need to supply two model dependent inputs: (1) the communication energy, that is the minimal energy barrier that the system must overcome to reach the stable state starting from the metastable one; (2) a recurrence property stating that for any configuration different from the metastable state there exists a path, starting from such a configuration and reaching a lower energy state, such that its maximal energy is lower than the communication energy.