BBS invariant measures with independent soliton components

Abstract

The Box-Ball System (BBS) is a one-dimensional cellular automaton in $\{0,1\}^\Z$ introduced by Takahashi and Satsuma \cite{TS}, who also identified conserved sequences called \emph{solitons}. Integers are called boxes and a ball configuration indicates the boxes occupied by balls. For each integer $k\ge1$, a $k$-soliton consists of $k$ boxes occupied by balls and $k$ empty boxes (not necessarily consecutive). Ferrari, Nguyen, Rolla and Wang \cite{FNRW} define the $k$-slots of a configuration as the places where $k$-solitons can be inserted. Labeling the $k$-slots with integer numbers, they define the $k$-component of a configuration as the array $\{\zeta_k(j)\}_{j\in \mathbb Z}$ of elements of $\Z_{\ge0}$ giving the number $\zeta_k(j)$ of $k$-solitons appended to $k$-slot $j\in \mathbb Z$. They also show that if the Palm transform of a translation invariant distribution $\mu$ has independent soliton components, then $\mu$ is invariant for the automaton. We show that for each $\lambda\in[0,1/2)$ the Palm transform of a product Bernoulli measure with parameter $\lambda$ has independent soliton components and that its $k$-component is a product measure of geometric random variables with parameter $1-q_k(\lambda)$, an explicit function of $\lambda$. The construction is used to describe a large family of invariant measures with independent components under the Palm transformation, including Markov measures.