Construction of soliton cellular automaton from the vertex model - The discrete 2D Toda equation and the Bogoyavlensky lattice

We study the soliton cellular automaton (SCA) in (2+1)-dimensions from the viewpoint of the integrable vertex model. As in our previous paper, we relate the SCA, the so-called box-ball system, to an integrable vertex model associated with the Bogoyavlensky lattice. We extend this framework and introduce the (2 + 1)-dimensional SCA, which can be interpreted as the ultradiscretization of the 2D Toda equation. We also construct the N-soliton solutions for this system.