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

Mathematical Physics & Probability

Speaker: Momar Dieng, UC Davis
Location: 693 Kerr
Start time: Tue, Oct 12 2004, 3:10PM

The distribution of the (properly scaled) largest eigenvalue of a p variate Wishart distribution on n degrees of freedom with identity covariance converges to the well-known F_1 Tracy-Widom distribution of RMT as n,p -> infinity with n/p = gamma >= 1. (Johnstone, 2001.) Equivalently, the result can be stated in terms of the square of the largest singular value of an n x p matrix X, all of whose entries are independent standard Gaussian variates, or the largest principal component of the covariance matrix X'X. This result is especially relevant to statisticians because of the explicit analytic form of the Tracy-Widom distributions. Building on the work of Tracy and Widom, we derive general analytic expressions for the distribution mth largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles (GOE, GSE) in terms of solutions to Painleve II. These are immediately relevant to the behavior of the mth largest eigenvalue of the appropriate Wishart distribution. In the process we also obtain an RMT proof of an interesting interlacing property between GOE and GSE eigenvalues scaled at the edge.