Abstract
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].
Highlights
Takahashi and Satsuma [7], referred to as TS in the sequel, introduced the Box-Ball System (BBS), a cellular automaton describing the deterministic evolution of a finite number of balls on the infinite lattice Z
We denote by T η the configuration obtained after the carrier has visited all boxes and T tη the configuration obtained after iterating this procedure t times, for positive integer t
Takahashi-Satsuma Identification of solitons We describe a variant of the TakahashiSatsuma algorithm [7] to identify the solitons of a finite ball configuration η
Summary
Takahashi and Satsuma [7], referred to as TS in the sequel, introduced the Box-Ball System (BBS), a cellular automaton describing the deterministic evolution of a finite number of balls on the infinite lattice Z. The soliton decomposition of an infinite configuration of balls is obtained applying the TS algorithm independently to each single finite excursion. Since the soliton decomposition is performed independently inside each excursion, it is convenient to introduce the finite array of components associated to one single excursion This combinatorial object is called a slot diagram. The components of an infinite configuration of balls is obtained suitably joining the slot diagrams of its excursions. For example (10) is the slot diagram associated to the excursion corresponding to the ball configuration in Fig. 1 and Fig. 5.
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