Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

TL;DR

This paper establishes an equivalence between spatial and temporal mixing conditions for certain probabilistic cellular automata, showing that exponential convergence to equilibrium is linked to decay of boundary influence, with implications for ergodicity and phase transitions.

Contribution

It proves the equivalence between spatial weak mixing and temporal ergodicity conditions for a class of PCA, connecting boundary influence decay to convergence rates.

Findings

Exponential ergodicity occurs when there is no phase transition.

Weak mixing implies exponential convergence to the Gibbs measure.

Boundary influence decay characterizes ergodic behavior.

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 inuence from the boundary for the invariant measures of the system restricted to nite boxes. For a class of reversible PCA dynamics on {--1, +1} Z d , with a naturally associated Gibbsian potential $\varphi$, we prove that a (spatial-) weak mixing condition (WM) for $\varphi$ implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to $\varphi$ holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.