Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

Abstract

For a general attractive Probabilistic Cellular Automata on S Z d , we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition (A). This condition means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite boxes. For a class of reversible PCA dynamics on {−1, +1} Z d , with a naturally associated Gibbsian potential ϕ, we prove that a (spatial-) weak mixing condition (WM) for ϕ implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to ϕ holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition. REPRINT,

Ergodicity of some classes of cellular automata subject to noise

“…The spatial mixing of the invariant measure is known for certain classes of ergodic PCA (see e.g. [39,55,45,43,10,6], as well as [59,47,31] in which the unique invariant measure is explicitly known). However, even the weaker condition of spatial ergodicity (≡ ergodicity under the shift action) is not known for general ergodic PCA.…”

“…For the class of PCA that are monotonic with respect to a total ordering of the alphabet, Louis [43] has provided a necessary and sufficient condition for exponential ergodicity in terms of a spatial mixing condition. Proposition 2.2 above established the computability of the unique invariant measure for every ergodic PCA.…”

Stability of the Uniqueness Regime for Ferromagnetic Glauber Dynamics Under Non-reversible Perturbations

“…Later, [23] showed that for translation-invariant attractive systems, the WSM condition (1.2) is in fact equivalent to (1.1). From our point of view, the main advantage of the WSM condition (1.2) is that we can show it to hold true throughout the uniqueness regime for ferromagnetic interactions.…”

Increasing coupling of probabilistic cellular automata

“…It was also used in Maes (1993). Recently, the coupling constructed in this paper was used to state some necessary and sufficient condition for the exponential ergodicity of attractive PCA's (see Louis (2004)). This last result relies on the fact that our coupling preserves a stochastic order between the configurations (so called increasing coupling).…”

“…We also give several examples and general applications of the constructed coupling. Indeed, the motivation for coupling together three or more PCA's comes from the paper Louis (2004), where a comparison between four different PCA dynamics proved to be useful.…”