Ergodicity versus non-ergodicity for Probabilistic Cellular Automata on\n rooted trees

2017/09/29 by Bruno Kimura, Kimura, Bruno, Wioletta M. Ruszel +3
Computer Science · Mathematics · Physics and Astronomy · #Cellular Automata and Applications #Cellular Automata and Lattice Gases (nlin.CG) #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1710.00084

openalex publication_date 2017/09/29 · openalex created_date 2019/12/13 · openalex updated_date 2026/07/28

Abstract

In this article we study a class of shift-invariant and positive rate\nprobabilistic cellular automata (PCA) on rooted d-regular trees mathbbT^d.\n In a first result we extend the results of [10] on trees, namely we prove\nthat to every stationary measure \ν of the PCA we can associate a space-time\nGibbs measure \μ_\ν on\n \ℤ \× mathbbT^d. Under certain assumptions on the dynamics\nthe converse is also true.\n A second result concerns proving sufficient conditions for ergodicity and\nnon-ergodicity of our PCA on d-ary trees for d\∈ 1,2,3 and\ncharacterizing the invariant product Bernoulli measures.\n

Citations

• Markovianity of the invariant distribution of probabilistic cellular automata on the line

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