Box-Ball System: Soliton and Tree Decomposition of Excursions

PABLO A. FERRARI; DAVIDE GABRIELLI

DF México

Simposio; XIII Symposium on Probability and Stochastic Processes; 2017

UNAM

Abstract We review combinatorial properties of solitons of the Box-Ball systemintroduced by Takahashi and Satsuma (J Phys Soc Jpn 59(10):3514?3519, 1990).Starting with several definitions of the system, we describe ways to identify solitonsand review a proof of the conservation of the solitons under the dynamics. Ferrariet al. (Soliton decomposition of the box-ball system (2018). arXiv:1806.02798)proposed a soliton decomposition of a configuration into a family of vectors, one foreach soliton size. Based on this decompositions, the authors (Ferrari and Gabrielli,Electron. J. Probab. 25, Paper No. 78?1, 2020) propose a family of measureson the set of excursions which induces invariant distributions for the Box-BallSystem. In the present paper, we propose a new soliton decomposition which isequivalent to a branch decomposition of the tree associated to the excursion, seeLe Gall (Une approche élémentaire des théorèmes de décomposition de Williams.In: Séminaire de Probabilités, XX, 1984/85, vol. 1204, pp. 447?464. LectureNotes in Mathematics. Springer, Berlin (1986)). A ball configuration distributed asindependent Bernoulli variables of parameter λ < 1/2 is in correspondence with asimple randomwalk with negative drift 2λ−1 and having infinitely many excursionsover the local minima. In this case the soliton decomposition of the walk consistson independent double-infinite vectors of iid geometric random variables (Ferrariand Gabrielli, Electron. J. Probab. 25, Paper No. 78?1, 2020). We show that thisproperty is shared by the branch decomposition of the excursion trees of the randomwalk and discuss a corresponding construction of a Geometric branching processwith independent but not identically distributed Geometric random variables.