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Seat number configuration of the box-ball system, and its relation to the 10-elimination and invariant measures

Hayate Suda

TL;DR

The paper develops a unified framework for linearizing BBS dynamics through seat-number configurations by introducing the $k$-skip map $\Psi_k$, which acts as a shift on seat numbers and generalizes the classical $10$-elimination. It extends these constructions from the half-line to the whole-line, establishing a time-linearization theorem with an offset and proving that the $k$-skip map preserves key invariants such as the $\zeta_k$-counts. By connecting the $k$-skip map to invariant measures in FG, the authors derive distributional results for $\Psi_k(\eta)$ under both excursion-based and Markovian measures, including explicit updates of parameters under $k$-skip and the resulting expectations of the seat-number carrier. The results provide a probabilistic lens on BBS linearization, with implications for randomized BBS and potential space-time scaling analyses of tagged solitons.

Abstract

The box-ball system (BBS) is a soliton cellular automaton introduced in [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 in [MSSS]. The aim of this paper is fourfold. First, we introduce the $k$-skip map $Ψ_{k} : Ω\to Ω$, where $Ω$ 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 in [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 $Ψ_{k}(η), η\in Ω$ when the distribution of $η$ belongs to a certain class of invariant measures of the BBS introduced in [FG]. As an application of the above results, we compute the expectation of the carrier with seat numbers.

Seat number configuration of the box-ball system, and its relation to the 10-elimination and invariant measures

TL;DR

The paper develops a unified framework for linearizing BBS dynamics through seat-number configurations by introducing the -skip map , which acts as a shift on seat numbers and generalizes the classical -elimination. It extends these constructions from the half-line to the whole-line, establishing a time-linearization theorem with an offset and proving that the -skip map preserves key invariants such as the -counts. By connecting the -skip map to invariant measures in FG, the authors derive distributional results for under both excursion-based and Markovian measures, including explicit updates of parameters under -skip and the resulting expectations of the seat-number carrier. The results provide a probabilistic lens on BBS linearization, with implications for randomized BBS and potential space-time scaling analyses of tagged solitons.

Abstract

The box-ball system (BBS) is a soliton cellular automaton introduced in [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 in [MSSS]. The aim of this paper is fourfold. First, we introduce the -skip map , where is the state space of the BBS, and show that the -skip map induces a shift operator of the seat number configuration. Second, we show that the -skip map is a natural generalization of the -elimination, which was originally introduced in [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 -skip map for the BBS on the whole-line. Finally, we investigate the distribution of when the distribution of belongs to a certain class of invariant measures of the BBS introduced in [FG]. As an application of the above results, we compute the expectation of the carrier with seat numbers.
Paper Structure (17 sections, 35 theorems, 219 equations, 3 figures)

This paper contains 17 sections, 35 theorems, 219 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Seat number configuration
  3. Carrier with seat numbers
  4. The k-skip map
  5. 10-elimination
  6. Definition of the 10-elimination
  7. On the relation to the seat number configuration
  8. BBS on the whole-line
  9. Seat number configuration on the whole-line
  10. Proof of Theorem \ref{['thm:linear_whole']}
  11. The k-skip map on the whole-line
  12. Relationships between k-skip map and invariant measures
  13. Excursion
  14. Distribution of k-skipped configuration
  15. Two-sided Markov distribution case
  16. ...and 2 more sections

Key Result

Lemma 2.3

Suppose that $\eta \in \Omega$. Then, for any $k \in {\mathbb{N}}$ and $x \in {\mathbb{N}}$, we have the following.

Figures (3)

  • Figure 1: Recursive construction of $\Phi_{k}(\eta)$ for $k = 1,2,3,4$.
  • Figure 2: Difference between the $1$-skip map and the $10$-elimination.
  • Figure 3: Identifying solitons in $\varepsilon$ by the TS Algorithm

Theorems & Definitions (76)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: Lemma 3.1 in MSSS
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6: Proposition 2.3 and Theorem 2.3 in MSSS
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 66 more