##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability and Statistics Seminar

## Brian Rider

#### University of Colorado, Boulder

## Spiking the Random Matrix Hard Edge

##### Abstract:

The largest eigenvalue of a rank one perturbation of random hermitian matrix is known to exhibit a phase transition. If the perturbation is small, one sees the famous Tracy-Widom law; if the perturbation is large, the result is simple Gaussian fluctuations. Further, there is a scaling window about a critical value of the perturbation which leads to a new one parameter family of limit laws. The same phenomena exists for random sample covariance matrices in which one of the population eigenvalues is "spiked", or takes a value other than one. Bloemendal-Virag have shown how this picture persists in the context of the general beta ensembles, giving new formulations of the discovered critical limit laws (among other things). Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to limit distributions. (Joint work with Jose Ramirez.)

Host: Todd Kemp

### February 2, 2012

### 9:00 AM

### AP&M 6402

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