Positive-rate PCA and IPS with stationary Bernoulli measures are rapidly forgetful

TL;DR

This paper proves that positive-rate probabilistic cellular automata with stationary Bernoulli measures exhibit exponential ergodicity and rapid mixing, with implications for their algorithmic distinguishability in higher dimensions.

Contribution

It establishes exponential ergodicity and logarithmic mixing times for a broad class of cellular automata and particle systems, using entropy-based methods.

Findings

Systems are exponentially ergodic with stationary Bernoulli measures.

Mixing time of finite regions is logarithmic in their diameter.

In higher dimensions, such automata are algorithmically indistinguishable from those without stationary Bernoulli measures.

Abstract

We prove that every probabilistic cellular automaton with strictly positive transition probabilities that admits a stationary Bernoulli measure is exponentially ergodic. Moreover, the mixing time of any finite region in such a system is logarithmic in the diameter of the region. A similar result holds in continuous time for positive-rate, finite-range interacting particle systems. The proofs use entropy, and rely on a representation of the system as a perturbation of another system with noise. The ergodic behaviour results from a competition between the accumulation of randomness due to noise and the diffusion of randomness due to local information exchange. We show that, in two and higher dimensions, the positive-rate probabilistic cellular automata that admit stationary Bernoulli measures are algorithmically indistinguishable from those that do not.