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Universality of the double scaling limit in random matrix models

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Tom Claeys, Arno B.J. Kuijlaars

Mathematical Physics · Submitted 31 Jan 2005

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

The authors' main claim We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit.

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arXiv IDmath-ph/0501074 (version 1)
Pith ID6ER34GIE, Pith's permanent record of this paper and its review
CitationArno B.J. Kuijlaars, Tom Claeys. "Universality of the double scaling limit in random matrix models." arXiv:math-ph/0501074 (2005). https://pith.science/paper/6ER34GIE
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,168 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.95 · mentions code or data availability 0.01.

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

We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painleve equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann-Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann-Hilbert problem is done with the Deift/Zhou steepest descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly.

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 study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. background 0.54 0.59 0.04 asserted 0.77
s1 We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. result 0.94 0.96 0.91 asserted 1.00
s2 The limiting kernels are constructed out of functions associated with the second Painleve equation. result 0.90 0.81 0.07 asserted 1.00
s3 This extends a result of Bleher and Its for the special case of a critical quartic potential. background 0.38 0.40 0.06 asserted 1.00
s4 The two main tools we use are equilibrium measures and Riemann-Hilbert problems. method 1.00 0.75 0.02 asserted 1.00
s5 In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. method 0.89 0.94 0.03 asserted 0.99
s6 The asymptotic analysis of the Riemann-Hilbert problem is done with the Deift/Zhou steepest descent analysis. method 1.00 0.81 0.02 asserted 1.00
s7 For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly. method 0.88 0.95 0.03 asserted 1.00

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

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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

The paper
Original abstract

We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painleve equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann-Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann-Hilbert problem is done with the Deift/Zhou steepest descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly.

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