Gaussian concentration bounds for probabilistic cellular automata

TL;DR

This paper investigates how Gaussian concentration bounds (GCB) evolve under probabilistic cellular automata (PCA), showing their conservation, conditions for stationary measures, and links to Gibbs measures.

Contribution

It establishes the preservation of GCB under PCA, characterizes GCB in high-noise regimes, and connects GCB properties with space-time Gibbs measures and automata contractiveness.

Findings

GCB is conserved under PCA dynamics.

GCB holds for the unique stationary measure in high-noise regimes.

GCB for space-time measures is equivalent to GCB for spatial marginals in contractive PCA.

Abstract

We study lattice spin systems and analyze the evolution of Gaussian concentration bounds (GCB) under the action of probabilistic cellular automata (PCA), which serve as discrete-time analogues of Markovian spin-flip dynamics. We establish the conservation of GCB and, in the high-noise regime, demonstrate that GCB holds for the unique stationary measure. Additionally, we prove the equivalence of GCB for the space-time measure and its spatial marginals in the case of contractive probabilistic cellular automata. Furthermore, we explore the relationship between (non)-uniqueness and GCB in the context of space-time Gibbs measures for PCA and illustrate these results with examples.