Bumping sequences and multispecies juggling

TL;DR

This paper generalizes juggling Markov chains to multispecies scenarios with balls of different weights, analyzing various models including add-drop, annihilation, and overwriting, and provides explicit formulas for stationary distributions.

Contribution

It introduces multispecies juggling models with bumping dynamics and derives explicit stationary distributions, including a novel overwriting model with ultra fast convergence.

Findings

Explicit stationary probability formulas derived for all models.

Introduction of a new overwriting model with ultra fast convergence.

Analysis of multiple jugglers exchanging balls in the models.

Abstract

Building on previous work by four of us (ABCN), we consider further generalizations of Warrington's juggling Markov chains. We first introduce "multispecies" juggling, which consist in having balls of different weights: when a ball is thrown it can possibly bump into a lighter ball that is then sent to a higher position, where it can in turn bump an even lighter ball, etc. We both study the case where the number of balls of each species is conserved and the case where the juggler sends back a ball of the species of its choice. In this latter case, we actually discuss three models: add-drop, annihilation and overwriting. The first two are generalisations of models presented in (ABCN) while the third one is new and its Markov chain has the ultra fast convergence property. We finally consider the case of several jugglers exchanging balls. In all models, we give explicit product formulas for the stationary probability and closed form expressions for the normalisation factor if known.