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We study the statistical properties of the long-time dynamics of the rule 54reversible cellular automaton (CA), driven stochastically at its boundaries.This CA can be considered as a discrete-time and deterministic version of theFredrickson-Andersen kinetically constrained model (KCM). By means of a matrixproduct ansatz, we compute the exact large deviation cumulant generatingfunctions for a wide range of time-extensive observables of the dynamics,together with their associated rate functions and conditioned long-timedistributions over configurations. We show that for all instances of boundarydriving the CA dynamics occurs at the point of phase coexistence betweencompeting active and inactive dynamical phases, similar to what happens in morestandard KCMs. We also find the exact finite size scaling behaviour of thesetrajectory transitions, and provide the explicit "Doob-transformed" dynamicsthat optimally realises rare dynamical events.

Exact large deviation statistics and trajectory phase transition of a deterministic boundary driven cellular automaton