Abstract

We study lattice spin systems and analyze the evolution of Gaussian concentration bounds (GCB) under the action of probabilistic cellular automata (PCA), which serve as discrete-time analogues of Markovian spin-flip dynamics. We establish the conservation of GCB and, in the high-noise regime, demonstrate that GCB holds for the unique stationary measure. Additionally, we prove the equivalence of GCB for the space-time measure and its spatial marginals in the case of contractive probabilistic cellular automata. Furthermore, we explore the relationship between (non)-uniqueness and GCB in the context of space-time Gibbs measures for PCA and illustrate these results with examples.

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The authors declare that they have no conflicts of interest to declare.

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Acknowledgements

The authors thank Christian Maes for useful advise on PCA’s and associated Gibbs measures. The authors also thank Pierre Collet for useful discussions.

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Authors and Affiliations

Centre de Physique Théorique, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, Palaiseau, France

Jean-René Chazottes 1.

Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

Frank Redig & Edgardo Ugalde 1.

Instituto de Física, Universidad Autónoma de San Luis Potosí, San Luis Potosí, México

Jean-René Chazottes, Frank Redig & Edgardo Ugalde

Authors

1. Jean-René Chazottes
2. Frank Redig
3. Edgardo Ugalde

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Correspondence to Edgardo Ugalde.

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Communicated by Aernout van Enter.

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Chazottes, JR., Redig, F. & Ugalde, E. Gaussian concentration bounds for probabilistic cellular automata. J Stat Phys 192, 169 (2025). https://doi.org/10.1007/s10955-025-03552-4

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Received: 09 July 2025

Accepted: 10 November 2025

Published: 21 November 2025

Version of record: 21 November 2025

DOI: https://doi.org/10.1007/s10955-025-03552-4