Exact critical threshold for percolation PCA (NDECA 254)

Determine the exact critical value of the mixture parameter λ at which the null diploid elementary cellular automaton with rule 254 (i.e., the probabilistic mixture of ECA rule 254 with the null rule) undergoes its inactive–active phase transition in the infinite-volume limit.

Background

The null diploid elementary cellular automaton (NDECA) with rule 254 is the probabilistic mixture of the null rule and ECA rule 254, widely known as the percolation probabilistic cellular automaton. Rigorous results establish the existence of a phase transition in this model, but the precise critical threshold in the mixture parameter λ that separates the absorbing phase from the active phase has not been determined exactly.

The literature provides rigorous bounds for this critical value—initially [1/3, 53/54], with subsequent improvements to the lower bound (0.505)—but the exact threshold remains undetermined. Accurately identifying this critical value would resolve a longstanding quantitative question about the percolation PCA’s phase transition and calibrate approximation methods such as Mean Field and block approximations.