Metastability for general dynamics with rare transitions: escape time and critical configurations

TL;DR

This paper extends the mathematical understanding of metastability and rare transitions in Markov chains beyond Metropolis systems, providing tools applicable to a broader class of stochastic models like Probabilistic Cellular Automata.

Contribution

It offers a general framework for analyzing metastability and critical configurations in non-Metropolis Markov chains, expanding the scope of previous energy landscape approaches.

Findings

Characterization of metastable states and transition times

Extension of energy landscape methods to non-Metropolis systems

Potential applications to Probabilistic Cellular Automata

Abstract

Metastability is a physical phenomenon ubiquitous in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions Markov chains. For Metropolis chains associated with Statistical Mechanics systems, this phenomenon has been described in an elegant way in terms of the energy landscape associated to the Hamiltonian of the system. In this paper, we provide a similar description in the general rare transitions setup. Beside their theoretical content, we believe that our results are a useful tool to approach metastability for non--Metropolis systems such as Probabilistic Cellular Automata.