Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations

We derive Painlevé-type expressions for the distribution of the mth largest eigenvalue in the Gaussian orthogonal and symplectic ensembles in the edge scaling limit. This work generalizes to general m the m=1 results of Tracy and Widom (1996). The results of Johnstone (2001) and Soshnikov (2002) imply the immediate relevance of our formulas for the mth largest eigenvalue of the appropriate Wishart distribution.