A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit

Benedikt Jahnel and Christof Külske

Stochastic Processes and their Applications, 2015, vol. 125, issue 6, 2427-2450

Abstract: We provide an example of a discrete-time Markov process on the three-dimensional infinite integer lattice with Zq-invariant Bernoulli-increments which has as local state space the cyclic group Zq. We show that the system has a unique invariant measure, but remarkably possesses an invariant set of measures on which the dynamics is conjugate to an irrational rotation on the continuous sphere S1. The update mechanism we construct is exponentially well localized on the lattice.

Keywords: Markov chain; Probabilistic cellular automaton; Interacting particle system; Non-equilibrium; Non-ergodicity; Rotation; Discretization; Gibbs measures; XY-model; Clock model (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:125:y:2015:i:6:p:2427-2450

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DOI: 10.1016/j.spa.2015.01.006

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