Distribution Functions for Edge Eigenvalues in Orthogonal and Symplectic Ensembles: Painlevé Representations II

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. This work generalizes to general $m$ the $m=1$ results of Tracy and Widom [23]. The results of Johnstone and Soshnikov (see [15], [19]) imply the immediate relevance of our formulas for the $m^{th}$ largest eigenvalue of the appropriate Wishart distribution.

V2: now includes Matlab(tm) code omitted in previous version; 146 pages, 2 figures

References in corpus (3)

• Level-Spacing Distributions and the Airy Kernel
• Inter-relationships between orthogonal, unitary and symplectic matrix ensembles
• Matrix Kernels for the Gaussian Orthogonal and Symplectic Ensembles

Cited by in corpus (3)

• On the Numerical Evaluation of Fredholm Determinants
• On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
• On the Evaluation of the Eigendecomposition of the Airy Integral Operator