On the spectral norm of Gaussian random matrices

TL;DR

This paper investigates the spectral norm of inhomogeneous Gaussian random matrices, confirming a conjecture up to a logarithmic factor and providing dimension-free bounds that enhance understanding of their geometric properties.

Contribution

It proves Latała's conjecture up to a  ext{log} ext{log} d factor and develops optimal dimension-free bounds for the spectral norm.

Findings

Spectral norm is of the same order as the largest row Euclidean norm, up to a  ext{log} ext{log} d factor.

Dimension-free bounds are established that are optimal to leading order.

Results illuminate the geometry of Gaussian processes underlying the matrices.

Abstract

Let $X$ be a $d\times d$ symmetric random matrix with independent but non-identically distributed Gaussian entries. It has been conjectured by Lata\l{a} that the spectral norm of $X$ is always of the same order as the largest Euclidean norm of its rows. A positive resolution of this conjecture would provide a sharp understanding of the probabilistic mechanisms that control the spectral norm of inhomogeneous Gaussian random matrices. This paper establishes the conjecture up to a dimensional factor of order $\sqrt{\log\log d}$. Moreover, dimension-free bounds are developed that are optimal to leading order and that establish the conjecture in special cases. The proofs of these results shed significant light on the geometry of the underlying Gaussian processes.