Behavior of known counterexamples at intermediate noise rates

Determine, for the existing one-dimensional counterexamples to the positive rates conjecture (notably Gács’s cellular automaton), whether the probabilistic cellular automata obtained by perturbation with uniform noise are ergodic at intermediate noise rates ε between the small-noise non-ergodic regime and the high-noise ergodic regime.

Background

The positive rates conjecture asserted that all one-dimensional probabilistic cellular automata (PCA) with strictly positive transition rates are ergodic; this was refuted by Gács via a one-dimensional cellular automaton that remains non-ergodic under sufficiently small uniform random noise. Conversely, for sufficiently high noise rates, any perturbed cellular automaton is known to be ergodic by percolation arguments.

For known counterexamples (such as Gács’s CA), the status for intermediate noise levels is not established. This paper constructs a new example exhibiting two phase transitions, but it does not resolve the behavior of the earlier counterexamples in the intermediate-noise regime.