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Scaling limits of solitons in the box-ball system

Stefano Olla, Makiko Sasada, Hayate Suda

Abstract

We study the space-time scaling limits of solitons in the box-ball system with random initial distribution. In particular, we show that any recentered tagged soliton converges to a Brownian motion in the diffusive space-time scale, and also prove the large deviation principle for the tagged soliton under certain shift-ergodic invariant distributions, including Bernoulli product measures and two-sided Markov distributions. Furthermore, in the diffusive space-time scaling, we show that two tagged solitons converge to the same Brownian motion even if they are macroscopically far apart.

Scaling limits of solitons in the box-ball system

Abstract

We study the space-time scaling limits of solitons in the box-ball system with random initial distribution. In particular, we show that any recentered tagged soliton converges to a Brownian motion in the diffusive space-time scale, and also prove the large deviation principle for the tagged soliton under certain shift-ergodic invariant distributions, including Bernoulli product measures and two-sided Markov distributions. Furthermore, in the diffusive space-time scaling, we show that two tagged solitons converge to the same Brownian motion even if they are macroscopically far apart.

Paper Structure

This paper contains 40 sections, 32 theorems, 389 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Box-Ball System
  3. Dynamics of the Box-Ball system
  4. Solitons in the Box-Ball configuration
  5. Set of all $k$-soltions
  6. Natural numbering for solitons
  7. Position of a soliton at time $n$
  8. Scaling limits for solitons
  9. Seat number configuration
  10. A soliton with volume
  11. Truncated numbering of solitons with volume
  12. Interactions between solitons
  13. Seat number configuration for the box-ball system
  14. General initial distributions
  15. q-statistics
  16. ...and 25 more sections

Key Result

Theorem 1

Let $X_{k}(n)$ be the position of the leftmost component of a tagged $k$-soliton at time $n$ and ${\mu}$ be a probability measure on $\{0,1\}^{{\mathbb{Z}}}$ satisfying the following. Assume that for some $k \in {\mathbb{N}}$, $k$-solitons exist with positive probability under ${\mu}$. Then, there exists some $v^{\mathrm{eff}}_{k} =v^{\mathrm{eff}}_k({\mu}) > 0$ such that

Figures (11)

  • Figure 1: $W$ and $T\eta$ obtained from $\eta =\dots 1100011100110010110000\dots$, where $\dots$ represents the consecutive $0$s.
  • Figure 2: Identifying solitons in $\eta$ by the TS Algorithm. $1$-soliton is colored by blue, $2$-solitons are colored by red, and $3$-soliton is colored by brown.
  • Figure 3: The $3$-soliton accelerates from time $2$ to $3$. On the other hand, the $1$-soliton does not move from time $1$ to $3$.
  • Figure 4: One $2$-soliton with volume $1$ and one $1$-soliton with volume $3$ are included in this figure. These solitons are interacting from the second line to fourth line. The $2$-soliton overtakes the group of $1$-solitons simultaneously.
  • Figure 5: One $2$-soliton with volume $2$ and one $1$-soliton with volume $1$ are included in this figure. These solitons are interacting from the second line to fifth line. Each $2$-solitons overtake the $1$-soliton step by step.
  • ...and 6 more figures

Theorems & Definitions (66)

  • Theorem : FNRW
  • Claim 1: Limit theorems for a tagged soliton
  • Remark 1.1
  • Claim 2: Strong correlations between $k$-solitons
  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem : Theorem 4.5 in S
  • ...and 56 more