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

[1806.02147] Dynamics of the box-ball system with random initial conditions via Pitman's transformation Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:1806.02147 (math) [Submitted on 6 Jun 2018] Title: Dynamics of the box-ball system with random initial conditions via Pitman's transformation Authors: David A. Croydon , Tsuyoshi Kato , Makiko Sasada , Satoshi Tsujimoto View a PDF of the paper titled Dynamics of the box-ball system with random initial conditions via Pitman's transformation, by David A. Croydon and 2 other authors View PDF HTML (experimental) Abstract: 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 the transformation of a nearest neighbour path encoding of the particle configuration given by `reflection in the past maximum', which was famously shown by Pitman to connect Brownian motion and a three-dimensional Bessel process. We use this to characterise the set of configurations for which the dynamics are well-defined and reversible for all times. We give simple sufficient conditions for random initial conditions to be invariant in distribution under the BBS dynamics, which we check in several natural examples, and also investigate the ergodicity of the relevant transformation. Furthermore, we analyse various probabilistic properties of the BBS that are commonly studied for interacting particle systems, such as the asymptotic behavior of the integrated current of particles and of a tagged particle. Finally, for Bernoulli product measures with parameter $p\uparrow\frac12$ (which may be considered the `high density' regime), the path encoding we consider has a natural scaling limit, which motivates the introduction of a new continuous version of the BBS that we believe will be of independent interest as a dynamical system. Subjects: Probability (math.PR) ; Mathematical Physics (math-ph) Cite as: arXiv:1806.02147 [math.PR] (or arXiv:1806.02147v1 [math.PR] for this version) https://doi.org/10.48550/arXiv.1806.02147 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Memoirs of the AMS, 283(1398)(2023) Related DOI : https://doi.org/10.1090/memo/1398 Focus to learn more DOI(s) linking to related resources Submission history From: David Croydon [ view email ] [v1] Wed, 6 Jun 2018 12:45:04 UTC (193 KB) Full-text links: Access Paper: View a PDF of the paper titled Dynamics of the box-ball system with random initial conditions via Pitman's transformation, by David A. Croydon and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2018-06 Change to browse by: math math-ph math.MP References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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