Abstract view
Fluctuation of matrix entries and application to outliers of elliptic matrices


Florent BenaychGeorges,
Université Paris Descartes, 45, rue des SaintsPères 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 SaintsPères 75270 Paris Cedex 06, France
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
random matrix, Gaussian fluctuation, spiked model, elliptic random matrix, Weingarten calculus, Haar measure
