Double Jump Phase Transition in a Soliton Cellular Automaton
Double Jump Phase Transition in a Soliton Cellular Automaton
复制标题
孤子元胞自动机中的双跳相变
DOI：
10.1093/imrn/rnaa166
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发表时间：
2017
影响因子：
1
通讯作者：
John Pike
中科院分区：
文献类型：
--
作者：
Lionel Levine;Hanbaek Lyu;John Pike
本文考虑了[26]中引入的具有随机初始配置的孤子元胞自动机。我们给出了杨氏图的多个结构，描述了系统的各种统计数据，根据熟悉的对象，如生与死链和高尔顿-沃森森林。利用这些思想，我们建立了极限定理，证明了如果1 $n$ 盒子被概率独立占据 $p\in (0,1)$，那么孤子的个数是有序的 $n$ 对所有人 $p$ 最长的孤子的长度是有序的 $\log n$ 为了 $p<1/2$，秩序 $\sqrt{n}$ 为了 $p=1/2$、秩序 $n$ 为了 $p>1/2$． 此外，我们还发现了超临界状态下的冷凝现象：对于每个固定 $j\geq 1$，顶部 $j$ 孤立子的长度与最长的顺序相同 $p\leq 1/2$，而除了最长的都有秩序 $\log n$ 为了 $p>1/2$． 作为一个应用，我们得到了的长度的缩放极限 $k^{\textrm{th}}$ 长度的随机堆栈排序排列中的最长递增和递减子序列 $n$ 用随机漫步和布朗漫游来表示。
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.