2025/07/04 by Marsan, Hugo, Sablik, Mathieu, Törmä, Ilkka · 3 citations
#37A50 #37B15 #60J05 #60K35 #Cellular Automata and Lattice Gases (nlin.CG) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR)
paper · doi:10.48550/arxiv.2507.03485
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 Gács, 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 Gács'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.