Fluctuation of Matrix Entries and Application to Outliers of Elliptic Matrices

Abstract

AbstractFor any family of N ⨯ N randommatrices that is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type Tr(AkM), where the matrix M is deterministic (such random variables include, for example, the normalized matrix entries of Ak). A consequence is the asymptotic independence of the projection of the matrices Ak 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 Formof 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.