# 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 centered i.i.d. entries with 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. In this paper, we study the singular value and singular vector statistics of Y_t around the right-most edge of the singular value spectrum in the harder regime n^-2/3≪ t ≪ 1. This regime is harder than the t=1 case, because on one hand, the edge behavior of the empirical spectral distribution (ESD) of YY^⊤ has a strong effect on the edge statistics of Y_tY_t^⊤ since t≪ 1 is "small", while on the other hand, the edge statistics of Y_t is also not merely a perturbation of those of Y since t≫ n^-2/3 is "large". Under certain regularity assumptions on Y, we prove the edge universality, eigenvalues rigidity and eigenvector delocalization for the matrices Y_tY_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 for Y_tY_t^⊤, and establish some sharp local laws on the resolvent of Y_tY_t^⊤. These results can be of independent interest, and used as useful inputs for many other problems regarding the spectral statistics of Y_t.

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