Seat number configuration of the box-ball system, and its relation to the 10-elimination and invariant measures
Seat number configuration of the box-ball system, and its relation to the 10-elimination and invariant measures
复制标题
箱球系统的座位数配置及其与10消元法和不变测度的关系
DOI：
--
复制
发表时间：
2023
期刊：
影响因子：
--
通讯作者：
Hayate Suda
中科院分区：
文献类型：
--
作者：
Hayate Suda
箱球系统（BBS）是由[TS]提出的一种孤立子元胞自动机，它的动力学可以通过几种方法线性化。最近，arXiv：2301.00132引入了一种新的线性化方法，称为座位数配置。本文的目的有四个方面。首先，我们引入了$k$-skip映射$\Psi_{k}：\Omega \to \Omega$，其中$\Omega$是BBS的状态空间，并证明了$k$-skip映射诱导了座位数配置的移位算子。第二，我们证明了$k$-skip映射是$10$-elimination的自然推广，它最初是由[MIT]引入的，用于解决周期BBS的初值问题。第三，我们推广了全线路BBS的座位数配置和$k$-跳跃映射的概念和结果。最后，当$\eta$的分布属于[FG]引入的BBS的某类不变测度时，我们研究了$\Psi_{k}（\eta），\eta \in \Omega$的分布.作为上述结果的应用，我们得到了具有马尔可夫平稳初始分布的积分电流的长时间行为.
The box-ball system (BBS) is a soliton cellular automaton introduced by [TS], and it is known that the dynamics of the BBS can be linearized by several methods. Recently, a new linearization method, called the seat number configuration, is introduced by arXiv:2301.00132. The aim of this paper is fourfold. First, we introduce the $k$-skip map $\Psi_{k} : \Omega \to \Omega$, where $\Omega$ is the state space of the BBS, and show that the $k$-skip map induces a shift operator of the seat number configuration. Second, we show that the $k$-skip map is a natural generalization of the $10$-elimination, which was originally introduced by [MIT] to solve the initial value problem of the periodic BBS. Third, we generalize the notions and results of the seat number configuration and the $k$-skip map for the BBS on the whole-line. Finally, we investigate the distribution of $\Psi_{k}(\eta), \eta \in \Omega$ when the distribution of $\eta$ belongs to a certain class of invariant measures of the BBS introduced by [FG]. As an application of the above results, we obtain the long-time behavior of the integrated current of $\Psi_{k}(\eta)$ with Markov stationary initial distributions.