Phase Transition in a Random Soliton Cellular Automaton

In this paper, we consider the soliton cellular automaton introduced in cite{takahashi1990soliton} 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 $pin(0,1)$, then the number of solitons is of order $n$ for all $p$, and the length of the longest soliton is of order $log n$ for $p 1/2$. Additionally, we uncover a condensation phenomenon in the supercritical regime: For each fixed $jgeq 1$, the top $j$ soliton lengths have the same order as the longest for $pleq 1/2$, whereas all but the longest have order at most $log n$ for $pu003e1/2$.