Dynamics of the Box-Ball System with Random Initial Conditions Via Pitman's Transformation / David A. Croydon, [and three others].

Format:

Book

Author/Creator:

Croydon, David A., author.

Series:

Memoirs of the American Mathematical Society ; Volume 283.

Memoirs of the American Mathematical Society Series ; Volume 283

Language:

English

Subjects (All):

Ergodic theory.

Cellular automata.

Pitman's measure of closeness.

Stochastic processes.

Topological dynamics.

Physical Description:

1 online resource (114 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"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 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 (i.e. can be inverted) for all times. The techniques developed to understand the deterministic dynamics are subsequently applied to study the evolution of the BBS from a random initial configuration. Specifically, 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 [up arrow] 1/2 (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"-- Provided by publisher.

Contents:

Path encodings of the BBS

Random initial configurations

Connections with Pitman's theorem and exclusion processes

BBS on R

Open questions

Notes:

Includes bibliographical references.

Description based on print version record.

Other Format:

Print version: Croydon, David A. Dynamics of the Box-Ball System with Random Initial Conditions Via Pitman's Transformation

ISBN:

9781470473990