Edge statistics of large dimensional deformed rectangular matrices

Abstract

We consider the edge statistics of large dimensional deformed rectangular matrices of the form Yt=Y+tX, 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 Yt 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 YY⊤ has a strong effect on the edge statistics of YtYt⊤ for a “small” t≪1, while on the other hand, the edge eigenvalue behavior of YtYt⊤ is not merely a perturbation of that of YY⊤ 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 YtYt⊤ and Yt⊤Yt. 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 YtYt⊤ 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 Yt.