dorsal/arxiv
View SchemaAiry Kernel and Painleve II
| Authors | Craig A. Tracy, Harold Widom |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9901004 |
| URL | https://arxiv.org/abs/solv-int/9901004 |
| Journal | in "Isomonodromic Deformations and Applications in Physics," eds. A. Its and J. Harnad, CRM Proceedings & Lecture Notes, Vol. 31, Amer. Math. Soc., Providence, 2002, pp. 85-98. |
Abstract
We prove that the distribution function of the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painlev\'e II. Our goal is to concentrate on this important example of the connection between random matrix theory and integrable systems, and in so doing to introduce the newcomer to the subject as a whole. We also give sketches of the results for the limiting distribution of the largest eigenvalue in the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble (GSE). This work we did some years ago in a more general setting. These notes, therefore, are not meant for experts in the field.
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"abstract": "We prove that the distribution function of the largest eigenvalue in the\nGaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in\nterms of Painlev\\\u0027e II. Our goal is to concentrate on this important example of\nthe connection between random matrix theory and integrable systems, and in so\ndoing to introduce the newcomer to the subject as a whole. We also give\nsketches of the results for the limiting distribution of the largest eigenvalue\nin the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble\n(GSE). This work we did some years ago in a more general setting. These notes,\ntherefore, are not meant for experts in the field.",
"arxiv_id": "solv-int/9901004",
"authors": [
"Craig A. Tracy",
"Harold Widom"
],
"categories": [
"solv-int",
"math-ph",
"math.MP",
"nlin.SI"
],
"journal_ref": "in \"Isomonodromic Deformations and Applications in Physics,\" eds.\n A. Its and J. Harnad, CRM Proceedings \u0026 Lecture Notes, Vol. 31, Amer. Math.\n Soc., Providence, 2002, pp. 85-98.",
"title": "Airy Kernel and Painleve II",
"url": "https://arxiv.org/abs/solv-int/9901004"
},
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