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Outlier Eigenvalues for Deformed I.I.D. Random Matrices
Author(s) -
Bordenave Charles,
Capitaine Mireille
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21629
Subject(s) - eigenvalues and eigenvectors , outlier , mathematics , random matrix , gaussian , complement (music) , distribution (mathematics) , measure (data warehouse) , limit (mathematics) , combinatorics , matrix (chemical analysis) , mathematical analysis , statistics , physics , computer science , quantum mechanics , biochemistry , chemistry , database , complementation , gene , phenotype , materials science , composite material
We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has i.i.d. entries with variance 1/ N . Under mild assumptions, as N grows the empirical distribution of the eigenvalues of A + Y converges weakly to a limit probability measure β on the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e., eigenvalues in the complement of the support of β . Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of Y or the Jordan decomposition of A . We also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function. © 2016 Wiley Periodicals, Inc.