# Dynamics of the box-ball system with random initial conditions via Pitman's transformation

@article{Croydon2018DynamicsOT, title={Dynamics of the box-ball system with random initial conditions via Pitman's transformation}, author={David A. Croydon and Tsuyoshi Kato and Makiko Sasada and Satoshi Tsujimoto}, journal={arXiv: Probability}, year={2018} }

The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al.\ recently showed the invariance in distribution of Bernoulli product measures with density strictly less than $\frac{1}{2}$, and gave a soliton decomposition for invariant measures more generally. We study the BBS dynamics using…

## 16 Citations

Dynamics of the multicolor box-ball system with random initial conditions via Pitman's transformation.

- Mathematics, Physics
- 2020

The Box-Ball System (BBS) is a cellular automaton introduced by Takahashi and Satsuma in the 1990s. The system is a discrete counterpart of the KdV equation and exhibits solitonic behavior. Recently,…

Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures

- Mathematics, PhysicsJournal of Mathematical Physics
- 2019

The box-ball system (BBS) is a simple model of soliton interaction introduced by Takahashi and Satsuma in the 1990s. Recent work of the authors, together with Tsuyoshi Kato and Satoshi Tsujimoto,…

Soliton Decomposition of the Box-Ball System

- Physics, MathematicsForum of Mathematics, Sigma
- 2021

Abstract The box-ball system (BBS) was introduced by Takahashi and Satsuma as a discrete counterpart of the Korteweg-de Vries equation. Both systems exhibit solitons whose shape and speed are…

Dynamics of the ultra-discrete Toda lattice via Pitman's transformation

- Physics, Mathematics
- 2019

By encoding configurations of the ultra-discrete Toda lattice by piecewise linear paths whose gradient alternates between $-1$ and $1$, we show that the dynamics of the system can be described in…

Box-Ball System: Soliton and Tree Decomposition of Excursions

- Mathematics, Physics
- 2020

We review combinatorial properties of solitons of the Box-Ball system introduced by Takahashi and Satsuma (J Phys Soc Jpn 59(10):3514–3519, 1990). Starting with several definitions of the system, we…

Double Jump Phase Transition in a Soliton Cellular Automaton

- Mathematics, Physics
- 2017

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…

Duality between box-ball systems of finite box and/or carrier capacity

- Mathematics, Physics
- 2019

We construct the dynamics of the box-ball system with box capacity $J$ and carrier capacity $K$, which we abbreviate to BBS($J$,$K$), in the case of infinite initial configurations, and show that…

Randomized box–ball systems, limit shape of rigged configurations and thermodynamic Bethe ansatz

- Physics, MathematicsNuclear Physics B
- 2018

Scaling limit of soliton lengths in a multicolor box-ball system

- Mathematics, Physics
- 2019

The box-ball systems are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de…

Discrete integrable systems and Pitman's transformation

- Mathematics, Physics
- 2020

We survey recent work that relates Pitman's transformation to a variety of classical integrable systems, including the box-ball system, the ultra-discrete and discrete KdV equations, and the…

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