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The relevance of IPS in statistical mechanics, as well as in many other contexts, is well established. Let us mention a couple of motivations for studying PCA. First, the investigation of fault-tolerant computational models was the motivation for the Russian school [13, 5]. Second, PCA appear in combinatorial problems related to the enumeration of directed animals [7]. An equi- librium is characterized by an invariant measure, that is a probability measure on the state space which is left invariant by the dynamic. By a compactness argument, there always exists at least one invariant measure.

There- fore, there are, a priori, three possible situations: 1 several invariant measures; 2 a unique invariant measure which is not attractive; 3 a unique invariant measure which is attractive. In the last case, which corresponds to the nicest possible situation, the model is said to be ergodic. Roughly, an ergodic system completely forgets about its initial condition, while a non-ergodic one remembers something forever. Date: November 30, Key words and phrases. Probabilistic cellular automaton; interacting particle system; ergodicity.

In other words, does uniqueness of the invariant measure imply convergence to it? For monotone systems, the intermediate case does not exist. But in general, the question is open. However, the question remains unsettled. For PCA, the same question is Unsolved problem 3.

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Therefore there is no convergence. This is consistent with the situation for Markov processes on a finite state space: in discrete time, periodic phenomena may occur which result in the existence of the intermediate case; in continuous time, the intermediate case does not exist.


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To prove the result for model A, we introduce two auxiliary PCA. We compute exactly the evolution of the one- dimensional marginals for model C Theorem 5.


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Continuous-time versions of models B and C have been studied in the IPS literature under the names of annihilating random walks and coalescing random walks, respectively, see [1, 3, 6]. Also, the coupling between the models B and C, that we use in Section 6, already appears in Griffeath [6, Ch. At last, let us mention that IPS versions of models B and C on a finite set of sites have also been studied, see for instance [2] for B, [9] for C, and the references therein.

Let us introduce probabilistic cellular automata, restricting ourselves to 1-dimensional models. Definition 2. The value of all the sites are updated.

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By specializing Definition 2. We obtain a deterministic cellular automaton. Thus we borrow the classical terminology of Markov chains. The PCA F is ergodic if it has a unique invariant measure which is attractive, i. Consider for a moment a Markov chain on a finite state space with transition matrix P. Let G P be the graph of the matrix P. In particular, uniqueness of the invariant measure does not imply ergodicity. For PCA, it was an open question to know if i implies ii in 1. The purpose of the present paper is to settle the question by proposing a non-ergodic PCA with a unique invariant measure.

Recall that each site behaves independently and as a finite Markov chain P. Let us justify the Table. Therefore, if P has several invariant measures, the same holds for F. Let us concentrate now on the intermediate case for P. Then F has an infinite number of invariant measures. Statement of the main result 3. Model A. A realization of the Markov chain is obtained as follows.

Invariant measure. The configuration 10 Z is defined similarly.

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Theorem 3. The PCA is non-ergodic. On configurations without 00 and 11, the PCA acts as the translation shift. Assume that it is the unique one. The purpose of Sections 4 and 5 is to prove Theorem 3. In model B, if two particles collide, then they annihilate. In model C, if two particles collide, they are merged into one particle.


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  • Let us define the models more formally. Model B. Let U be an update process, defined as in Section 3. The transition function of model B.

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    In the above presentation, model B is a Markov chain with synchronous updates and local interactions, but not stricto sensu a PCA. Indeed, if Y0 is deterministic, then the r.

    If you know the book but cannot find it on AbeBooks, we can automatically search for it on your behalf as new inventory is added. If it is added to AbeBooks by one of our member booksellers, we will notify you! Griffeath ; David Griffeath. Publisher: Springer , This specific ISBN edition is currently not available. Bernoulli 19 , no. Abstract Article info and citation First page References Abstract The style of mathematical models known to probabilists as Interacting Particle Systems and exemplified by the Voter, Exclusion and Contact processes have found use in many academic disciplines.

    Article information Source Bernoulli , Volume 19, Number 4 , Export citation. Export Cancel. References [1] Aldous, D.