# Randomly juggling backwards

- Published in 2016
- Added on

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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.

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### BibTeX entry

@article{Randomlyjugglingbackwards, title = {Randomly juggling backwards}, 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.}, url = {http://arxiv.org/abs/1601.06391v1 http://arxiv.org/pdf/1601.06391v1}, author = {Allen Knutson}, comment = {}, urldate = {2017-06-14}, archivePrefix = {arXiv}, eprint = {1601.06391}, primaryClass = {math.CO}, year = 2016, collections = {attention-grabbing-titles,probability-and-statistics} }