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A Perturbed Cellular Automaton with Two Phase Transitions for the Ergodicity



Tekijät: Marsan, Hugo; Sablik, Mathieu; Törmä, Ilkka

Julkaisuvuosi: 2026

Lehti: Journal of Statistical Physics

Artikkelin numero: 6

Vuosikerta: 193

Numero: 1

ISSN: 0022-4715

eISSN: 1572-9613

DOI: https://doi.org/10.1007/s10955-025-03564-0

Julkaisun avoimuus kirjaamishetkellä: Avoimesti saatavilla

Julkaisukanavan avoimuus : Osittain avoin julkaisukanava

Verkko-osoite: https://doi.org/10.1007/s10955-025-03564-0

Rinnakkaistallenteen osoite: https://research.utu.fi/converis/portal/detail/Publication/508988842

Rinnakkaistallenteen lisenssi: CC BY

Rinnakkaistallennetun julkaisun versio: Kustantajan versio


Tiivistelmä

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.


Ladattava julkaisu

This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail. Please cite the original version.

Julkaisussa olevat rahoitustiedot: 
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).


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