Distribution Functions for Edge Eigenvalues in Orthogonal and Symplectic Ensembles: Painlevé Representations
Abstract
We derive Painlevé--type expressions for the distribution of the
m
th
largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the edge scaling limit. The work of Johnstone and Soshnikov (see [7], [10]) implies the immediate relevance of our formulas for the
m
th
largest eigenvalue of the appropriate Wishart distribution.