A perturbed cellular automaton with two phase transitions for the ergodicity

Hugo Marsan, Mathieu Sablik, Ilkka Törmä

correctmedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper rigorously constructs a one-dimensional CA T with an auxiliary arrow/counter layer and proves a double phase transition in ergodicity: small-noise and high-noise uniform ergodicity, and a specific intermediate noise value with non-ergodicity (Theorem 3.1). Its intermediate-noise proof carefully reduces to a conditional product bound (Theorem 2.1) derived from Gács, and its low-noise ergodicity is shown via a Markov additive chain analysis of dependence cones. In contrast, the candidate solution’s intermediate-noise argument incorrectly appeals to an unsubstantiated “density threshold” for Gács-style reliability and uses a domination-by-i.i.d. heuristic that does not imply the required conditional product bound; its small-noise percolation-style renormalization also leaves key independence/separation details unstated.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The work establishes a novel double phase transition phenomenon for one-dimensional PCA and executes the proof with a careful blend of Gács-inspired reliability and a fresh Markov additive chain analysis for dependence cones. While technically long, the arguments are coherent and well-motivated. Minor expository refinements could improve readability and portability.