Bumping sequences and multispecies juggling

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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.

MSC

60C05
60J10
05A05
82C23

Keywords

Markov chains
Combinatorics
Juggling

Cited by (0)

The first author (A.A.) acknowledges support from a UGC Centre for Advanced Study grant and Department of Science and Technology grant DST/INT/SWD/VR/P-01/2014, and thanks LIAFA for hospitality during his stay there, where this work was initiated. J.B. and S.C. acknowledge financial support from the Agence Nationale de la Recherche via the grants ANR-08-JCJC-0011 “IComb”, ANR 12-JS02-001-01 “Cartaplus” and ANR-14-CE25-0014 “GRAAL”, and from the “Combinatoire à Paris” project funded by the City of Paris. S.L. acknowledges support from the Swedish Research Council, grant 621-2014-4780. F.N. acknowledges support from the Raman-Charpak Fellowship programme.