Largest eigenvalue of positive mean Gaussian matrices

TL;DR

This paper investigates the fluctuations of the largest eigenvalue in symmetric Gaussian matrices with positive mean and correlated entries, showing it converges to a normal distribution under certain conditions, extending previous results.

Contribution

It generalizes known results for Wigner matrices by analyzing correlated Gaussian matrices with positive mean, providing explicit normal convergence conditions.

Findings

Largest eigenvalue, after centering, converges to a normal distribution.

Convergence holds under the assumption of an absolutely summable covariance kernel.

Results extend classical findings for matrices with independent entries.

Abstract

This short note studies the fluctuations of the largest eigenvalue of symmetric random matrices with correlated Gaussian entries having positive mean. Under the assumption that the covariance kernel is absolutely summable, it is proved that the largest eigenvalue, after centering, converges in distribution to normal with an explicitly defined mean and variance. This result generalizes known findings for Wigner matrices with independent entries.