Probabilistic cellular automata, invariant measures, and perfect sampling

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.

💡 Research Summary

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The paper treats probabilistic cellular automata (PCA) as infinite‑dimensional Markov chains whose transition rule is a local probability distribution depending only on a finite neighbourhood. After formalising this viewpoint, the authors focus on two central questions: (i) when does a PCA possess a unique invariant measure to which all initial distributions converge (ergodicity), and (ii) how can one obtain exact samples from that invariant measure.

The first major result concerns one‑dimensional cellular automata (the deterministic special case of PCA). The authors prove that for any 1‑D CA the property of ergodicity is equivalent to nilpotency – the existence of a finite time after which every configuration evolves to the same uniform configuration (typically the all‑zero state). Since nilpotency is known to be an undecidable property (it subsumes the halting problem), the paper concludes that deciding ergodicity for 1‑D CA (and therefore for the corresponding deterministic PCA) is also undecidable. This establishes a sharp theoretical limitation: no algorithm can, in general, determine whether a given 1‑D CA will converge to a unique stationary distribution.

The second contribution is an algorithm for perfect sampling of the invariant measure of an ergodic PCA. Classical perfect‑sampling methods such as Coupling from the Past (CFTP) rely on monotonicity or a partial order on the state space, which many interesting PCA lack. To overcome this, the authors introduce a “bounding process”. The bounding process is itself a PCA that evolves two extremal configurations – a maximal and a minimal one – using the same local transition probabilities. Both extremal configurations are updated synchronously; when they coalesce (become identical) at some past time (-T), all possible initial configurations must have merged into the same state at time 0. Consequently, the state at time 0 obtained from either extremal trajectory is an exact sample from the stationary distribution. The key insight is that the bounding process inherits the PCA structure, so the algorithm can be implemented with the same simulation engine used for the original model, and it does not require any monotonicity assumption. The algorithm proceeds by running the bounding process backwards from a sufficiently large negative time until coalescence occurs, then returning the coalesced state as a perfect sample.

The third part of the paper applies this method to the “PCA Majority” rule, a well‑studied but analytically intractable example. In Majority PCA each cell updates to the majority value among itself and its two neighbours, with a probability that can be tuned. The long‑term behaviour of this rule (whether it exhibits a unique stationary measure, multiple phases, or complex mixtures) has remained unknown. Using the perfect‑sampling algorithm, the authors generate exact samples for a wide range of parameter values and compute empirical statistics such as the average density of 1’s, spatial correlation functions, and cluster‑size distributions. Their experiments reveal a phase‑like transition: for high majority‑bias the system tends to the all‑1 configuration, for low bias it favours all‑0, while intermediate parameters produce a mixture of large domains and persistent fluctuations. These observations suggest that Majority PCA may possess multiple invariant measures or a non‑trivial ergodic component, phenomena that cannot be captured by monotone coupling arguments.

Finally, the paper discusses future directions. Extending the bounding‑process technique to non‑PCA Markov chains, analysing coalescence times theoretically, and investigating undecidability results in higher dimensions are highlighted as promising avenues. In summary, the work (1) establishes the undecidability of ergodicity for 1‑D cellular automata via a nilpotency equivalence, (2) introduces a novel, monotonicity‑free perfect‑sampling algorithm based on a bounding PCA, and (3) leverages this algorithm to shed new light on the elusive dynamics of the Majority PCA. These contributions deepen our theoretical understanding of stochastic cellular automata and provide a practical tool for exact simulation of their stationary behaviour.