Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions
Pierre-Yves Louis, Technische Universitat Berlin
Abstract
For a general attractive Probabilistic
Cellular Automata on SZ^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.
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