  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. In this
article, we survey the invariant measures that were presented in the latter
work, 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 the previous work was to describe scaling limits for the box-ball
system; here, we review these results, 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 ultra-discrete Toda lattice.
