Double Jump Phase Transition in a Soliton Cellular Automaton

Abstract

Abstract In this paper, we consider the soliton cellular automaton introduced in [ 26] 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 1st $n$ boxes are occupied independently with probability $p\in (0,1)$, 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 $p<1/2$, order $\sqrt{n}$ for $p=1/2$, and order $n$ for $p>1/2$. Additionally, we uncover a condensation phenomenon in the supercritical regime: for each fixed $j\geq 1$, the top $j$ soliton lengths have the same order as the longest for $p\leq 1/2$, whereas all but the longest have order $\log n$ for $p>1/2$. As an application, we obtain scaling limits for the lengths of the $k^{\textrm{th}}$ longest increasing and decreasing subsequences in a random stack-sortable permutation of length $n$ in terms of random walks and Brownian excursions.