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Random Unitary Matrices, Permutations and Painleve

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Craig A. Tracy, Harold Widom

Combinatorics · Submitted 25 Nov 1998

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

The authors' main claim Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painleve II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral group.

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arXiv IDmath/9811154 (version 2)
Pith ID7YIW52NK, Pith's permanent record of this paper and its review
CitationCraig A. Tracy, Harold Widom. "Random Unitary Matrices, Permutations and Painleve." arXiv:math/9811154 (1998). https://pith.science/paper/7YIW52NK
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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, 2,468 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.03 · states quantitative result 0.15 · compares with prior results 0.33 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

This paper is concerned with certain connections between the ensemble of n x n unitary matrices -- specifically the characteristic function of the random variable tr(U) -- and combinatorics -- specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups -- and the appearance of Painleve functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painleve II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral group.

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 This paper is concerned with certain connections between the ensemble of n x n unitary matrices -- specifically the characteristic function of the random variable tr(U) -- and combinatorics -- specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups -- and the appearance of Painleve functions in the answers to apparently unrelated questions. background 0.54 0.24 0.21 asserted 0.96
s1 Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painleve II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral group. result 0.99 0.80 0.49 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

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  • 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

This paper is concerned with certain connections between the ensemble of n x n unitary matrices -- specifically the characteristic function of the random variable tr(U) -- and combinatorics -- specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups -- and the appearance of Painleve functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painleve II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral group.

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2 arXiv papers cite this in 4 sentences

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

Papers citing it · 2 sentences without a confident label

Uses its method or tool 1 paper
  • To study the (non)perturbative corrections to the free energy, it is convenient to use the method of orthogonal polynomial $p_n(z)$ obeying [equation] The partition function of GWW model is written in terms of the norm $h_n$ as [equation] From the constant term $f_n$ of $p_n(z)$ [Note that we have shifted the index $n$ of $f_n$ by one as compared to the definition of [ref] .] [equation] we can compute the ratio of the norm $h_n$ [equation] From [ref] and [ref] one can show that [equation] Furthermore, using the recursion relation [equation] one can show that $f_n$ satisfies [equation] Note that this is known as a discrete Painlevé equation [ref] .

    Wilson loops in unitary matrix models at finite $N$ · Instanton correction in the ungapped phase · 0.93

Cites it as background 1 paper
  • The Gross-Witten model is related to the Painlevé II equation, an important result in the physics literature [ref] that is now known to be a particular case of a more general mathematical relationship between the matrix model and the theory of Painlevé transcendents [ref] .

    Two-dimensional Yang-Mills theory, Painleve equations and the six-vertex model · Antiferroelectric phase and Chern-Simons theory on $S^3/ℤ_2$ · 0.85

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