Double Jump Phase Transition in a Random Soliton Cellular Automaton

In this paper, we consider the soliton cellular automaton introduced in [21] with a random initial configuration. We give multiple constructions of a Young diagram describing various statistics of the system in terms of familiar objects like birth-and-death chains and Galton-Watson forests. Using these ideas, we establish limit theorems showing that if the first n boxes are occupied independently with probability p ∈ (0,1), then the number of solitons is of order n for all p, and the length of the longest soliton is of order logn for p < 1/2, order pn for p = 1/2, and order n for p > 1/2. Additionally, we uncover a condensation phenomenon in the supercritical regime: For each fixed j ≥ 1, the top j soliton lengths have the same order as the longest for p ≤ 1/2, whereas all but the longest have order at most logn for p > 1/2. As an application, we obtain scaling limits for the lengths of the kth longest increasing and decreasing subsequences in a random stacksortable permutation of length n in terms of random walks and Brownian excursions.