Abstract
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. The normalization factor in one case can be explicitly written as a homogeneous symmetric polynomial. We also refine and generalize enriched Markov chains on set partitions. Lastly, we prove that in one case, the stationary distribution is attained in bounded time.
Highlights
Juggling as a human endeavour has been around since time immemorial, it is fairly recently that mathematicians have taken an active interest in exploring the field
Multivariate juggling probabilities mathematics appear for instance in algebraic geometry [8, 14]
The simplest model considered by Warrington, where the number of balls is conserved, has been generalized by Leskelä and Varpanen as the so-called Juggler’s Exclusion Process (JEP), where the balls can be thrown arbitrarily high so that the state space is infinite [16]
Summary
Juggling as a human endeavour has been around since time immemorial, it is fairly recently that mathematicians have taken an active interest in exploring the field. The simplest model considered by Warrington, where the number of balls is conserved, has been generalized by Leskelä and Varpanen as the so-called Juggler’s Exclusion Process (JEP), where the balls can be thrown arbitrarily high so that the state space is infinite [16]. Extensions to a fluctuating number of balls (but with a finite state space) are considered in Section 4: we provide the multivariate extension of the add-drop and the annihilation models introduced in [22], in the respective Sections 4.1 and 4.2. These models have the same transition graph, only the transitions probabilities differ. The claims of this paper can be verified by downloading the MapleTM program RandomJuggling either from the arXiv source or the first author’s (A.A.) homepage
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
