Rare-event properties in a classical stochastic model describing the evolution of random unitary circuits

We investigate the statistics of selected rare events in a (1+1)-dimensional (classical) stochastic growth model which describes the evolution of (quantum) random unitary circuits. In such classical formulation, particles are created and/or annihilated at each step of the evolution process, according to rules which generally favor a growing cluster size. We apply a large-deviation approach based on biased Monte Carlo simulations, with suitable adaptations, to evaluate (a) the probability of ending up with a single particle at a specified final time \(t_f\), and (b) the probability of having particles outside the light cone, defined by a "butterfly velocity" \(v_B\), at \(t_f\). Morphological features of single-particle final configurations are discussed, in connection with whether the location of such particle is inside or outside the light cone; we find that joint occurrence of both events of types (a) and (b) drives significant changes to such features, signalling a second-order phase transition.