Tracy-widom law for the extreme eigenvalues of large signal-plus-noise matrices

• Main Authors: Zhang, Zhixiang, Liu, Yiming, Pan, Guangming
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2024
• Subjects: Physics; Edge universality; Extreme eigenvalues
• Online Access: https://hdl.handle.net/10356/173676
• Institution: Nanyang Technological University

Let S = R + X be an M × N matrix where R is the signal matrix and X is the noise matrix consisting of i.i.d. standardized entries. The signal matrix R is allowed to be full rank, which is rarely studied in literature compared with the low rank cases. Under a regularity condition of R that assures the square root behaviour of the spectral density near the edge, we prove that the largest eigenvalue of SS* has the Tracy-Widom distribution under a tail condition on the entries of X. Moreover, such a condition is proved to be necessary and sufficient to assure the Tracy-Widom law.