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

Abstract

Let $\mathbf{S}=\mathbf{R}+\mathbf{X}$ be an $M\times N$ 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 $\mathbf{S}{\mathbf{S}}^{\ast }$ 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.