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# Fluctuation of matrix entries and application to outliers of elliptic matrices

Published:2017-11-15

• Florent Benaych-Georges,
Université Paris Descartes, 45, rue des Saints-Pères 75270 Paris Cedex 06, France
• Guillaume Cébron,
IMT, Université Paul Sabatier, 118 Route de Narbonne 31062 Toulouse Cedex 04, France
• Jean Rochet,
Université Paris Descartes, 45, rue des Saints-Pères 75270 Paris Cedex 06, France
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## Abstract

For any family of $N\times N$ random matrices $(\mathbf{A}_k)_{k\in K}$ which is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type $\operatorname{Tr}(\mathbf{A}_k \mathbf{M})$, where the matrix $\mathbf{M}$ is deterministic (such random variables include for example the normalized matrix entries of the $\mathbf{A}_k$'s). A consequence is the asymptotic independence of the projection of the matrices $\mathbf{A}_k$ onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. These results are used to study the asymptotic behavior of the outliers of a spiked elliptic random matrix. More precisely, we show that the fluctuations of these outliers around their limits can have various rates of convergence, depending on the Jordan Canonical Form of the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other. These phenomena have already been observed with random matrices from the Single Ring Theorem.
 Keywords: random matrix, Gaussian fluctuation, spiked model, elliptic random matrix, Weingarten calculus, Haar measure
 MSC Classifications: 60B20 - Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 15B52 - Random matrices 60F05 - Central limit and other weak theorems 46L54 - Free probability and free operator algebras

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