Randomly juggling backwards

TL;DR

This paper explores a novel random walk on the reverse graph of juggling states, linking it to classical linear algebra and infinite-dimensional geometry, and analyzes its equilibrium distribution and asymptotic behavior.

Contribution

It introduces a new random walk model on the reverse juggling graph and connects it to the Poincaré series of the b-Grassmannian, providing insights into its equilibrium and asymptotics.

Findings

Derived a simple Boltzmann distribution as equilibrium

Identified the most likely asymptotic state with many balls

Linked the equilibrium to the Poincaré series of the b-Grassmannian

Abstract

We recall the directed graph of _juggling states_, closed walks within which give juggling patterns, as studied by Ron Graham in [w/Chung, w/Butler]. Various random walks in this graph have been studied before by several authors, and their equilibrium distributions computed. We motivate a random walk on the reverse graph (and an enrichment thereof) from a very classical linear algebra problem, leading to a particularly simple equilibrium: a Boltzmann distribution closely related to the Poincar\'e series of the b-Grassmannian in infinite-dimensional space. We determine the most likely asymptotic state in the limit of many balls, where in the limit the probability of a 0-throw is kept fixed.