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.Comment: 14 pages, 1 figure. References updated in second versio
