##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability and Statistics Seminar

## Brian Rider

#### University of Colorado at Boulder

## Beta Ensembles, Random Schroedinger, and Diffusion

##### Abstract:

The beta ensembles of random matrix theory are natural generalizations of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles, these classical cases corresponding to beta = 1, 2, or 4. We prove that the largest eigenvalues in the general ensembles have limit laws described by the low lying spectrum of certain random Schroedinger operators, providing a new characterization of the celebrated Tracy-Widom laws. As a corollary, a second characterization is available via the explosion probability of an associated one-dimensional diffusion. A complementary picture is developed for beta versions of random sample-covariance matrices. (Based on work with J. Ramirez and B. Virag.)

Host: Jason Schweinsberg

### December 4, 2008

### 9:00 AM

### AP&M 6402

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