I will discuss recent work with Lionel Levine and Hanbaek Lyu concerning the behavior of the soliton cellular automaton with random initial conditions. This CA is a discrete-time dynamical system which models the behavior of certain traveling wave packets arising in various areas of math and physics. After explaining the basics of the model, I will describe connections with a variety of combinatorial objects like pattern-avoiding permutations, Young tableaux, and Motzkin paths. I will then turn to the random setting where one can frame things in terms of probabilistic constructs like renewal processes, birth-and-death chains, Brownian motions, and Galton-Watson forests. Using these perspectives, I will talk about some limit theorems which establish a 'double jump phase transition' for certain statistics of the system analogous to that found by Erdős and Rényi in their seminal study of random graphs.