Spectral norm of matrices with independent entries up to polyloglog

TL;DR

This paper establishes tight bounds on the spectral norm of random matrices with independent entries, showing that the operator norm is controlled by explicit quantities and supporting a conjecture about its behavior.

Contribution

It provides a new upper bound on the spectral norm of matrices with independent entries under moment growth conditions, refining previous estimates with a smaller iterated logarithm factor.

Findings

Derived explicit upper bounds for the spectral norm

Provided two-sided estimates up to a small iterated logarithm factor

Supported a conjecture on the behavior of spectral norms of such matrices

Abstract

In this paper, we study the expectation of the operator norm of the random matrix (a_{ij} X_{ij}) for i,j <= n, under the assumption that the random variables (X_{ij}) are independent, symmetric and satisfy the moment growth condition ||X_{ij}||{2p} <= C ||X_{ij}||{p} for every p >= 1. We derive an upper bound expressed in terms of quantities that can be explicitly computed in many cases. This bound implies a two-sided estimate, up to a factor given by a power of an iterated logarithm. This factor is considerably smaller than the natural scale of the problem. Our result thus provides positive evidence supporting a conjecture formulated by Rafal Latala and Jan Swiatkowski.