A Path-Counting Analysis of Phase Shifts in Box-Ball Systems

TL;DR

This paper provides a rigorous, path-counting proof of the phase shift formula in the box-ball system, linking it to the Toda lattice and introducing a new discrete-time analogue.

Contribution

It introduces a novel proof technique using path-counting and connects the phase shift phenomena of the box-ball system with the Toda lattice.

Findings

Derived a new explicit formula for phase shifts in the box-ball system.

Established a connection between the box-ball system and the Toda lattice.

Provided a discrete-time Toda lattice analogue of the phase shift phenomenon.

Abstract

In this paper, we perform a detailed analysis of the phase shift phenomenon of the classical soliton cellular automaton known as the box-ball system, ultimately resulting in a statement and proof of a formula describing this phase shift. This phenomenon has been observed since the nineties, when the system was first introduced by Takahashi and Satsuma, but no explicit global description was made beyond its observation. By using the Gessel-Viennot-Lindstr\"om lemma and path-counting arguments, we present here a novel proof of the classical phase shift formula for the continuous-time Toda lattice, as discovered by Moser, and use this proof to derive a discrete-time Toda lattice analogue of the phase shift phenomenon. By carefully analysing the connection between the box-ball system and the discrete-time Toda lattice, through the mechanism of tropicalisation/dequantisation, we translate this discrete-time Toda lattice phase shift formula into our new formula for the box-ball system phase shift.