Fluctuation of matrix entries and application to outliers of elliptic matrices
Résumé
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 conditions for
central limit theorems of random variables of the type
$\operatorname{Tr}(\mathbf{A}_k \mathbf{M})$, where the Euclidean norm of
$\mathbf{M}$ has order $\sqrt{N}$ (such random variables include for example
the normalized matrix entries $\sqrt{N} \mathbf{A}_k(i,j)$). 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. This result is used to study the asymptotic
behaviour of the outliers of a spiked elliptic random matrix. More precisely,
we show that their fluctuations 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 by Benaych-Georges and Rochet with random matrices from the Single
Ring Theorem.