The Ghost-Box-Ball System: A Unified Perspective on Soliton Cellular Automata, the RSK Algorithm and Phase Shifts

TL;DR

This paper introduces the ghost-box-ball system, extending classical soliton cellular automata, and explores its deep connections with the RSK algorithm, phase shifts, and solitonic phenomena, providing new insights into integrable dynamical systems.

Contribution

It presents the ghost-box-ball system as a unified framework linking soliton cellular automata, the RSK algorithm, and phase shifts, expanding classical notions and analyzing their interrelations.

Findings

Established the ghost-box-ball system as a generalization of the classical box-ball system.

Uncovered the relationship between the ghost-box-ball system and the RSK correspondence.

Analyzed the ultradiscrete phase shift phenomenon within the new framework.

Abstract

In this paper, we introduce the ghost-box-ball system, which is an extended version of the classical soliton cellular automaton. It is initially motivated as a mechanism for making precise a connection between the Schensted insertion (of the Robinson-Schensted-Knuth correspondence) and the dynamical process of the box-ball system. In addition to this motivation, we explore generalisations of classical notions of the box-ball system, including the solitonic phenomenon, the asymptotic sorting property, and the invariant shape construction. We analyse the ghost-box-ball system beyond its initial relevance to the Robinson-Schensted-Knuth correspondence, unpacking its relationship to its underlying dynamical evolution on a coordinatisation and using a mechanism for augmenting a regular box-ball configuration to study the classical ultradiscrete phase shift phenomenon.