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

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.