In this paper, we consider the soliton cellular automaton introduced in [Takahashi 1990] 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 $p\in(0,1)$, 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 $p 1/2$. 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 at most $\log n$ for $p>1/2$. As an application, we obtain scaling limits for the lengths of the $k^{\text{th}}$ longest increasing and decreasing subsequences in a random stack-sortable permutation of length $n$ in terms of random walks and Brownian excursions.

中文翻译：

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孤子元胞自动机中的双跳相变

在本文中，我们考虑在 [Takahashi 1990] 中引入的具有随机初始配置的孤子元胞自动机。我们给出了杨图的多种构造，根据熟悉的对象（如生死链和高尔顿-沃森森林）描述系统的各种统计数据。使用这些想法，我们建立了极限定理，表明如果前 $n$ 个盒子以概率 $p\in(0,1)$ 被独立占据，那么对于所有 $p$，孤子的数量为 $n$，并且最长孤子的长度为 $\log n$ 为 $p 1/2$。此外，我们发现了超临界区的凝聚现象：对于每个固定的 $j\geq 1$，顶部的 $j$ 孤子长度与 $p\leq 1/2$ 的最长孤子长度具有相同的顺序，而除了最长的订单最多 $\log n$ 为 $p>1/2$。作为应用程序，