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

## TL;DR

This paper provides an exact analysis of the large deviation statistics and phase transition behavior in a boundary-driven deterministic cellular automaton, revealing coexistence of active and inactive phases and explicit dynamics for rare events.

## Contribution

It introduces an exact matrix product approach to compute large deviation functions and characterizes phase coexistence in a deterministic CA with boundary driving.

## Key findings

- Phase coexistence between active and inactive dynamical phases
- Exact finite size scaling of trajectory transitions
- Explicit Doob-transformed dynamics for rare events

## Abstract

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