BBS invariant measures with independent soliton components

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].