Zhigang Bao: Phase transition for the smallest eigenvalue of covariance matrices

Speaker
Zhigang Bao, University of Hong Kong

Abstract
In studies of extreme eigenvalues of Wigner matrices and the largest eigenvalue of sample covariance matrices, it has been established that a weak 4th moment condition is necessary and sufficient for the Tracy-Widom law to hold. In this talk, we will show that the Tracy-Widom law is more robust for the smallest non-zero eigenvalue of the sample covariance matrix. We will specifically illustrate a phase transition from the Tracy-Widom distribution to a Gaussian distribution when the tail exponent of the matrix entry distribution crosses 8/3. If time permits, we will also discuss a phase transition for the localization length of the bottom eigenvector when the tail exponent crosses 2. This talk is based on a joint work with Jaehun Lee and Xiaocong Xu.