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Randomly juggling backwards

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Allen Knutson

Combinatorics · Submitted 24 Jan 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

how it is stated · asserted 1.00 read the record →

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arXiv ID1601.06391 (version 1)
Pith IDORIS4JYJ, Pith's permanent record of this paper and its review
CitationAllen Knutson. "Randomly juggling backwards." arXiv:1601.06391 (2016). https://pith.science/paper/ORIS4JYJ
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,670 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.11 · human participants 0.03 · controlled comparison 0.03 · randomized assignment 0.03 · states quantitative result 0.26 · compares with prior results 0.39 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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]. background 1.00 0.05 0.03 asserted 0.98
s1 Various random walks in this graph have been studied before by several authors, and their equilibrium distributions computed. background 1.00 0.03 0.02 asserted 1.00
s2 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. contribution 0.91 0.91 0.45 asserted 1.00
s3 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. result 0.93 0.93 0.39 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original 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.

Human review

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 5 sentences without a confident label

Compares with 2 papers
Cites as background 6 papers
  • In [ref] are studied Markov chains of juggling states, in which the possible throws from a state are given probabilities.

    Juggler's exclusion process · A Markov chain · 0.96

  • In [ref] are studied Markov chains of juggling states, in which the possible throws from a state are given probabilities.

    Geometric juggling with q-analogues · A Markov chain · 0.96

  • In [ref] are studied Markov chains of juggling states, in which the possible throws from a state are given probabilities.

    Multivariate Juggling Probabilities · A Markov chain · 0.96

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