Edge statistics of large dimensional deformed rectangular matrices.

We consider the edge statistics of large dimensional deformed rectangular matrices of the form Y t = Y + t X , where Y is a p × n deterministic signal matrix whose rank is comparable to n , X is a p × n random noise matrix with i.i.d. entries of mean zero and variance n − 1 , and t > 0 gives the noise level. This model is referred to as the interference-plus-noise matrix in the study of massive multiple-input multiple-output (MIMO) system, which belongs to the category of the so-called signal-plus-noise model. For the case t = 1 , the spectral statistics of this model have been studied to a certain extent in the literature (Dozier and Silverstein, 2007[17,18]; Vallet et al., 2012). In this paper, we study the singular value and singular vector statistics of Y t around the right-most edge of the spectrum in the harder regime n − 1 / 3 ≪ t ≪ 1. This regime is harder than the t = 1 case, because on the one hand, the edge behavior of the empirical spectral distribution (ESD) of Y Y ⊤ has a strong effect on the edge statistics of Y t Y t ⊤ for a "small" t ≪ 1 , while on the other hand, the edge eigenvalue behavior of Y t Y t ⊤ is not merely a perturbation of that of Y Y ⊤ for a "large" t ≫ n − 1 / 3 . Under certain regularity assumptions on Y , we prove the Tracy–Widom law for the edge eigenvalues, the eigenvalue rigidity, and eigenvector delocalization for the matrices Y t Y t ⊤ and Y t ⊤ Y t . These results can be used to estimate and infer the massive MIMO system. To prove the main results, we analyze the edge behavior of the asymptotic ESD of Y t Y t ⊤ and establish optimal local laws on its resolvent. These results are of independent interest, and can be used as important inputs for many other problems regarding the spectral statistics of Y t. [ABSTRACT FROM AUTHOR]