Outliers in the Single Ring Theorem

• 成果类型: Article
• 署名作者: Benaych-Georges, Florent; Rochet, Jean
• 署名单位: Universite Paris Cite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0632-x
• 发表日期: 2016
• 页码: 313-363
• 关键词: largest eigenvalue; rank perturbations; deformations; CONVERGENCE; unitary

摘要

This text is about spiked models of non-Hermitian random matrices. More specifically, we consider matrices of the type , where the rank of stays bounded as the dimension goes to infinity and where the matrix is a non-Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if has some eigenvalues out of the maximal circle of the single ring, then has some eigenvalues (called outliers) in the neighborhood of those of , which is not the case for the eigenvalues of in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of around the eigenvalues of and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such kind of fluctuations had already been shown for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of ) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in (Ann Probab 42:1980-2031, 2014) in the Hermitian case, but only for non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a result by Tao proved specifically for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new.