Double Jump Phase Transition in a Soliton Cellular Automaton. (29th July 2020)

• Record Type: Journal Article
• Authors: Levine, Lionel; Lyu, Hanbaek; Pike, John
• Is Part Of: International mathematics research notices. Volume 2022:Number 1(2022)
• Page Start: 665
• Page End: 727
• Publication Date: 2020-07-29
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa166
• Languages: English
• ISSNs: 1073-7928

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.