The spectrum of coupled random matrices

Mark Adler; Pierre van Moerbeke

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The spectrum of coupled random matrices

Mark Adler, Pierre van Moerbeke

High Energy Physics - Theory · Submitted 29 Jul 1999

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

The authors' main claim These methods lead to very simple nonlinear third-order partial differential equations for the joint statistics of the spectra of two coupled Gaussian random matrices.

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What the abstract itself states, as probabilities: claims first 0.16 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.02 · states quantitative result 0.15 · compares with prior results 0.28 · mentions code or data availability 0.02.

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

The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this work, we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled Hermitean random matrices and the equations the associated probabilities satisfy. In particular, the two-Toda lattice, its algebra of symmetries and its vertex operators will play a prominent role in this interaction. Namely, the method is to introduce time parameters, in an artificial way, and to dress up a certain matrix integral with a vertex integral operator, for which we find Virasoro-like differential equations. These methods lead to very simple nonlinear third-order partial differential equations for the joint statistics of the spectra of two coupled Gaussian random matrices.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The study of the spectrum of coupled random matrices has received rather little attention.	background 0.51	0.03	0.02	asserted 0.99
s1	To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta.	background 0.98	0.04	0.02	asserted 0.60
s2	In this work, we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled Hermitean random matrices and the equations the associated probabilities satisfy.	contribution 0.99	0.81	0.21	asserted 0.92
s3	In particular, the two-Toda lattice, its algebra of symmetries and its vertex operators will play a prominent role in this interaction.	method 0.75	0.29	0.06	asserted 0.94
s4	Namely, the method is to introduce time parameters, in an artificial way, and to dress up a certain matrix integral with a vertex integral operator, for which we find Virasoro-like differential equations.	method 1.00	0.89	0.08	asserted 1.00
s5	These methods lead to very simple nonlinear third-order partial differential equations for the joint statistics of the spectra of two coupled Gaussian random matrices.	result 0.99	0.92	0.62	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.78 · evidence from proof or derivation 1.00.

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arXiv listing abstract v2

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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

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Original abstract

The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this work, we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled Hermitean random matrices and the equations the associated probabilities satisfy. In particular, the two-Toda lattice, its algebra of symmetries and its vertex operators will play a prominent role in this interaction. Namely, the method is to introduce time parameters, in an artificial way, and to dress up a certain matrix integral with a vertex integral operator, for which we find Virasoro-like differential equations. These methods lead to very simple nonlinear third-order partial differential equations for the joint statistics of the spectra of two coupled Gaussian random matrices.

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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

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