Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures

Abstract

The box-ball system (BBS) is a simple model of soliton interaction introduced by Takahashi and Satsuma in the 1990s. Recent work of the authors, together with Tsuyoshi Kato and Satoshi Tsujimoto, derived various families of invariant measures for the BBS based on two-sided stationary Markov chains [D. A. Croydon et al., “Dynamics of the box-ball system with random initial conditions via Pitman’s transformation,” arXiv:1806.02147]. In this article, we survey the invariant measures that were presented in D. A. Croydon et al. and also introduce a family of new ones for periodic configurations that are expressed in terms of Gibbs measures. Moreover, we show that the former examples can be obtained as infinite volume limits of the latter. Another aspect of D. A. Croydon et al. was to describe scaling limits for the BBS; here, we review the results of D. A. Croydon et al. and also present scaling limits other than those that were covered there. One, the zigzag process has previously been observed in the context of queuing; another, a periodic version of the zigzag process, is apparently novel. Furthermore, we demonstrate that certain Palm measures associated with the stationary and periodic versions of the zigzag process yield natural invariant measures for the dynamics of corresponding versions of the ultradiscrete Toda lattice.