• Corpus ID: 119131097

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… 

Figures from this paper

Dynamics of the multicolor box-ball system with random initial conditions via Pitman's transformation.
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
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
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
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
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
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
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
Scaling limit of soliton lengths in a multicolor box-ball system
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
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
...
1
2
...

References

SHOWING 1-10 OF 42 REFERENCES
Phase Transition in a Random Soliton Cellular Automaton
In this paper, we consider the soliton cellular automaton introduced in \cite{takahashi1990soliton} with a random initial configuration. We give multiple constructions of a Young diagram describing
On a periodic soliton cellular automaton
We propose a box and ball system with a periodic boundary condition periodic box and ball system (pBBS). The time evolution rule of the pBBS is represented as a Boolean recurrence formula, an inverse
Integrable structure of box–ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry
The box–ball system is an integrable cellular automaton on a one-dimensional lattice. It arises from either quantum or classical integrable systems by procedures called crystallization and
Markov Chains and Mixing Times
This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary
An Elementary Proof of the Hitting Time Theorem
In this note, we give an elementary proof of the random walk hitting time theorem, which states that, for a left-continuous random walk on Z starting at a nonnegative integer k, the conditional
LETTER TO THE EDITOR: Box and ball system with a carrier and ultradiscrete modified KdV equation
A new soliton cellular automaton is proposed. It is defined by an array of an infinite number of boxes, a finite number of balls and a carrier of balls. Moreover, it reduces to a discrete equation
Continuous martingales and Brownian motion
0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.-
...
1
2
3
4
5
...