In this talk, I will introduce infinite versions of four well-studied discrete integrable models, namely
the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and
the discrete Toda equation. These systems are understood as "deterministic vertex model”, which are discretely indexed in space and time, and their
deterministic dynamics is defined locally via lattice equations. They have another formulation via the generalized Pitman’s transform, which is a new crucial observation.
We show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support
of many shift ergodic measures. Also, a detailed balance criterion is presented that, amongst the measures that describe spatially independent and identically/
alternately distributed configurations, characterizes those that are temporally invariant in
distribution. This talk is based on a joint work with David Croydon and Satoshi Tsujimoto.