Interesting Esoterica

Randomly juggling backwards

Article by Allen Knutson
  • Published in 2016
  • Added on
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}
}