Introductory Workshop Random Matrix Models and Their Applications Airy Kernel and Painlevé Ii
نویسندگان
چکیده
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 Painlevé 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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Higher order analogues of the Tracy-Widom distribution and the Painlevé II hierarchy
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