Airy Kernel and Painleve II

Details Airy Kernel and Painleve II Airy Kernel and Painleve II

1999-01-15

arxiv.org/abs/solv-int/9901004v2

in "Isomonodromic Deformations and Applications in Physics," eds. A. Its and J. Harnad, CRM Proceedings & Lecture Notes, Vol.

Physics

Mathematical Physics

14 pages, 1 figure. References updated in second version

Scientific paper

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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