Probabilistic cellular automata, invariant measures, and perfect sampling

TL;DR

This paper studies the ergodicity of probabilistic cellular automata (PCA), proves undecidability of ergodicity in 1D, and introduces a perfect sampling algorithm for invariant measures, with applications to the PCA Majority.

Contribution

It establishes the equivalence of ergodicity and nilpotency in 1D cellular automata and develops a novel perfect sampling method for ergodic PCA without monotonicity assumptions.

Findings

Ergodicity in 1D cellular automata is undecidable.

A new perfect sampling algorithm for invariant measures of ergodic PCA is proposed.

Numerical experiments on the PCA Majority demonstrate the algorithm's applicability.

Abstract

A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a 1-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to be also a PCA. Last, we focus on the PCA Majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.