A1 Refereed original research article in a scientific journal
A Perturbed Cellular Automaton with Two Phase Transitions for the Ergodicity
Authors: Marsan, Hugo; Sablik, Mathieu; Törmä, Ilkka
Publisher: Springer Science+Business Media
Publication year: 2026
Journal: Journal of Statistical Physics
Article number: 6
Volume: 193
Issue: 1
ISSN: 0022-4715
eISSN: 1572-9613
DOI: https://doi.org/10.1007/s10955-025-03564-0
Publication's open availability at the time of reporting: Open Access
Publication channel's open availability : Partially Open Access publication channel
Web address : https://doi.org/10.1007/s10955-025-03564-0
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/508988842
Self-archived copy's licence: CC BY
Self-archived copy's version: Publisher`s PDF
The positive rates conjecture states that a one-dimensional probabilistic cellular automaton (PCA) with strictly positive transition rates must be ergodic. The conjecture has been refuted by Gács, whose counterexample is a cellular automaton that is non-ergodic under uniform random noise with sufficiently small rate. For all known counterexamples, non-ergodicity has been proved under small enough rates. Conversely, all cellular automata are ergodic with sufficiently high-rate noise. No other types of phase transitions of ergodicity are known, and the behavior of known counterexamples under intermediate noise rates is unknown. We present an example of a cellular automaton with two phase transitions. Using Gács’s result as a black box, we construct a cellular automaton that is ergodic under small noise rates, non-ergodic for slightly higher rates, and again ergodic for rates close to 1.
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Funding information in the publication:
Open Access funding provided by University of Turku (including Turku University Central Hospital). Ilkka Törmä was supported by the Academy of Finland grant 359921, and a visiting researcher grant from Paul Sabatier University. M. Sablik acknowledges the support of the ANR “Difference” project (ANR-20-CE40-0002).
