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2020 BBS invariant measures with independent soliton components
Pablo A. Ferrari, Davide Gabrielli
Electron. J. Probab. 25: 1-26 (2020). DOI: 10.1214/20-EJP475


The Box-Ball System (BBS) is a one-dimensional cellular automaton in the configuration space $\{0,1\}^{\mathbb{Z} }$ introduced by Takahashi and Satsuma [8], who identified conserved quantities called solitons. Ferrari, Nguyen, Rolla and Wang [4] map a configuration to a family of soliton components, indexed by the soliton sizes $k\ge 1$. Building over this decomposition, we give an explicit construction of a large family of invariant measures for the BBS that are also shift invariant, including Ising-like Markov and Bernoulli product measures. The construction is based on the concatenation of iid excursions of the associated walk trajectory. Each excursion has the property that the law of its $k$ component given the larger components is product of a finite number of geometric distributions with a parameter depending on $k$. As a consequence, the law of each component of the resulting ball configuration is product of identically distributed geometric random variables, and the components are independent. This last property implies invariance for BBS, as shown by [4].


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Pablo A. Ferrari. Davide Gabrielli. "BBS invariant measures with independent soliton components." Electron. J. Probab. 25 1 - 26, 2020.


Received: 6 December 2018; Accepted: 8 June 2020; Published: 2020
First available in Project Euclid: 11 July 2020

zbMATH: 07252710
MathSciNet: MR4125783
Digital Object Identifier: 10.1214/20-EJP475

Primary: 37B15, 37K40, 60C05


Vol.25 • 2020
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