Let $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $ where $ \bbX_n $ is a $ p \times n $ matrix with independent standardized random variables, $ \bbR_n $ is a $ p \times n $ non-random matrix, representing the information, and $ \bbT_{n} $ is a $ p \times p $ non-random nonnegative definite Hermitian matrix. Under some conditions on $ \bbR_n \bbR_n^* $ and $ \bbT_n $, it has been proved that for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all $ p $ sufficiently large. The purpose of this paper is to carry on with the study of the support of the limit spectral distribution, and we show that there is an exact separation phenomenon: with probability one, the proper number of eigenvalues lie on either side of these intervals.

翻译：设 $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $，其中 $ \bbX_n $ 为 $ p \times n $ 矩阵，元素为独立标准化随机变量；$ \bbR_n $ 为 $ p \times n $ 非随机矩阵，表示信息部分；$ \bbT_{n} $ 为 $ p \times p $ 非随机非负定 Hermite 矩阵。在 $ \bbR_n \bbR_n^* $ 和 $ \bbT_n $ 满足一定条件下，已有证明：对于极限谱分布支撑集外的任意闭区间，当 $ p $ 足够大时，几乎必然没有特征值落入该区间。本文旨在进一步研究极限谱分布的支撑集，并揭示一种精确分离现象：几乎必然地，在这些区间两侧恰有相应数量的特征值分布。