A perturbed cellular automaton with two phase transitions for the ergodicity

TL;DR

This paper introduces a cellular automaton that exhibits two phase transitions in ergodicity as noise rates vary, filling a gap in understanding the complex behavior of probabilistic cellular automata.

Contribution

The authors construct a cellular automaton with two distinct ergodicity phase transitions, demonstrating complex behavior not previously observed in such systems.

Findings

Ergodic at low noise rates

Non-ergodic at intermediate noise rates

Re-ergodic at high noise rates

Abstract

The positive rates conjecture states that a one-dimensional probabilistic cellular automaton (PCA) with strictly positive transition rates must be ergodic. The conjecture has been refuted by G\'acs, whose counterexample is a cellular automaton that is non-ergodic under uniform random noise with sufficiently small rate. For all known counterexamples, non-ergodicity has been proved under small enough rates. Conversely, all cellular automata are ergodic with sufficiently high-rate noise. No other types of phase transitions of ergodicity are known, and the behavior of known counterexamples under intermediate noise rates is unknown. We present an example of a cellular automaton with two phase transitions. Using G\'acs's result as a black box, we construct a cellular automaton that is ergodic under small noise rates, non-ergodic for slightly higher rates, and again ergodic for rates close to 1.