# Box-ball system: soliton and tree decomposition of excursions

## TL;DR

This paper reviews the combinatorial properties of solitons in the Box-Ball system, introduces a new tree-based soliton decomposition, and explores its probabilistic implications for random walk excursions and geometric branching processes.

## Contribution

It proposes a novel soliton decomposition based on tree branch decomposition, linking combinatorial and probabilistic structures in the Box-Ball system.

## Key findings

- Soliton decomposition corresponds to a branch decomposition of excursion trees.
- Random walk excursions with Bernoulli-distributed configurations have independent geometric soliton vectors.
- The branch decomposition shares properties with the soliton decomposition, leading to a geometric branching process.

## Abstract

We review combinatorial properties of solitons of the Box-Ball system introduced by Takahashi and Satsuma in 1990. Starting with several definitions of the system, we describe ways to identify solitons and review a proof of the conservation of the solitons under the dynamics. Ferrari, Nguyen, Rolla and Wang 2018 proposed a soliton decomposition of a configuration into a family of vectors, one for each soliton size. Based on this decompositions, the authors have proposed a family of measures on the set of excursions which induces invariant distributions for the Box-Ball System. In this paper, we propose a new soliton decomposition which is equivalent to a branch decomposition of the tree associated to the excursion, see Le Gall 2005. A ball configuration distributed as independent Bernoulli variables of parameter $\lambda<1/2$ is in correspondence with a simple random walk with negative drift $2\lambda-1$ and infinitely many excursions over the local minima. In this case the authors have proven that the soliton decomposition of the walk consists on independent double-infinite vectors of iid geometric random variables. We show that this property is shared by the branch decomposition of the excursion trees of the random walk and discuss a corresponding construction of a Geometric branching process with independent but not identically distributed Geometric random variables.