Fluctuation of matrix entries and application to outliers of elliptic matrices

TL;DR

This paper establishes central limit theorems for trace-related variables of invariant random matrices and applies these results to analyze the fluctuations and correlations of outliers in spiked elliptic matrices.

Contribution

It provides general conditions for CLTs of trace functionals of invariant matrices and explores their application to outlier behavior in spiked elliptic models, including fluctuation rates and correlations.

Findings

Outliers can fluctuate at different rates depending on the Jordan form.

Asymptotic independence of trace projections onto null trace subspace.

Possible correlations between distant outliers in elliptic matrices.

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 by Benaych-Georges and Rochet with random matrices from the Single Ring Theorem.