Electronic Journal of Probability

Multivariate juggling probabilities

Arvind Ayyer, Jérémie Bouttier, Sylvie Corteel, and François Nunzi

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We consider refined versions of Markov chains related to juggling introduced by Warrington. We further generalize the construction to juggling with arbitrary heights as well as infinitely many balls, which are expressed more succinctly in terms of Markov chains on integer partitions. In all cases, we give explicit product formulas for the stationary probabilities and closed-form expressions for the normalization factor. We also refine and generalize enriched Markov chains on set partitions. Lastly, we prove that in one case, the stationary distribution is attained in finite time.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 5, 29 pp.

Accepted: 15 January 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 05A17: Partitions of integers [See also 11P81, 11P82, 11P83] 05A18: Partitions of sets 82C23: Exactly solvable dynamic models [See also 37K60]

Markov chain Combinatorics Juggling

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Ayyer, Arvind; Bouttier, Jérémie; Corteel, Sylvie; Nunzi, François. Multivariate juggling probabilities. Electron. J. Probab. 20 (2015), paper no. 5, 29 pp. doi:10.1214/EJP.v20-3495. https://projecteuclid.org/euclid.ejp/1465067111

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  • Arita, Chikashi; Bouttier, Jérémie; Krapivsky, P. L.; Mallick; Kirone. Asymmetric exclusion process with global hopping. Phys. Rev. E 88 (2013), 042120. arXiv:1307.4367
  • Ayyer, Arvind; Schilling, Anne; Steinberg, Benjamin; Thiéry, Nicolas M. Markov chains, $\mathscr{R}$-trivial monoids and representation theory. Int. J. Algebra Comput., to appear. arXiv:1401.4250
  • Ayyer, Arvind; Strehl, Volker. Stationary distribution and eigenvalues for a de Bruijn process. Advances in Combinatorics (Ilias S. Kotsireas and Eugene V. Zima, eds.), Springer Berlin Heidelberg, 2013, pp. 101–120 (English). arXiv:1108.5695
  • Buhler, Joe; Eisenbud, David; Graham, Ron; Wright, Colin. Juggling drops and descents. Amer. Math. Monthly 101 (1994), no. 6, 507–519.
  • Butler, Steve; Graham, Ron. Enumerating (multiplex) juggling sequences. Ann. Comb. 13 (2010), no. 4, 413–424.
  • Chung, Fan; Graham, Ron. Universal juggling cycles. Combinatorial number theory, 121–130, de Gruyter, Berlin, 2007.
  • Chung, Fan; Graham, Ron. Primitive juggling sequences. Amer. Math. Monthly 115 (2008), no. 3, 185–194.
  • Devadoss, Satyan L.; Mugno, John. Juggling braids and links. Math. Intelligencer 29 (2007), no. 3, 15–22.
  • Ehrenborg, Richard; Readdy, Margaret. Juggling and applications to $q$-analogues. Proceedings of the 6th Conference on Formal Power Series and Algebraic Combinatorics (New Brunswick, NJ, 1994). Discrete Math. 157 (1996), no. 1-3, 107–125.
  • Engström, Alexander; Leskelä, Lasse; Varpanen, Harri. Geometric juggling with q-analogues, 2013. arXiv:1310.2725
  • Evans, M. R.; Hanney, T. Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A 38 (2005), no. 19, R195–R240.
  • Evans, M. R.; Majumdar, Satya N.; Zia, R. K. P. Factorized steady states in mass transport models on an arbitrary graph. J. Phys. A 39 (2006), no. 18, 4859–4873.
  • Gould, H. W. The $q$-Stirling numbers of first and second kinds. Duke Math. J. 28 1961 281–289.
  • Knutson, Allen; Lam, Thomas; Speyer, David E. Positroid varieties: juggling and geometry. Compos. Math. 149 (2013), no. 10, 1710–1752.
  • Le Gall, J.-F. Intégration, probabilités et processus aléatoires, ENS lectures notes, 2006, available at http://www.math.u-psud.fr/~jflegall/
  • Leskelä, Lasse; Varpanen, Harri. Juggler's exclusion process. J. Appl. Probab. 49 (2012), no. 1, 266–279.
  • Levin, David A.; Peres, Yuval; Wilmer, Elizabeth L. Markov chains and mixing times. With a chapter by James G. Propp and David B. Wilson. American Mathematical Society, Providence, RI, 2009. xviii+371 pp. ISBN: 978-0-8218-4739-8
  • Polster, Burkard. The mathematics of juggling. Springer-Verlag, New York, 2003. xviii+226 pp. ISBN: 0-387-95513-5
  • Sagan, Bruce E. A maj statistic for set partitions. European J. Combin. 12 (1991), no. 1, 69–79.
  • Stadler, Jonathan Derek. Schur functions, juggling, and statistics on shuffled permutations. Thesis (Ph.D.)-The Ohio State University. ProQuest LLC, Ann Arbor, MI, 1997. 82 pp. ISBN: 978-0591-51450-6
  • Stadler, Jonathan D. Juggling and vector compositions. Discrete Math. 258 (2002), no. 1-3, 179–191.
  • Warrington, Gregory S. Juggling probabilities. Amer. Math. Monthly 112 (2005), no. 2, 105–118.