Multivariate Juggling Probabilities

TL;DR

This paper extends Markov chain models of juggling to more complex scenarios, providing explicit formulas for stationary distributions and analyzing convergence times, thus advancing the mathematical understanding of juggling probabilities.

Contribution

It introduces generalized Markov chain models for juggling with arbitrary heights and infinitely many balls, offering explicit stationary probability formulas and convergence analysis.

Findings

Explicit product formulas for stationary probabilities

Normalization factor as a homogeneous symmetric polynomial

Stationary distribution is reached in bounded time in one case

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.