On the operator norm of a Hermitian random matrix with correlated entries

TL;DR

This paper analyzes the operator norm of a correlated Hermitian random matrix with polynomial decay in correlations, demonstrating it is stochastically dominated by one through moment calculations and cumulant decay.

Contribution

It introduces a method to bound the operator norm of correlated Hermitian matrices with polynomial decay in correlations, extending previous results to more general dependence structures.

Findings

Operator norm is stochastically dominated by one.

Moment calculations confirm the boundedness of the operator norm.

Polynomial decay in correlations ensures the validity of the bounds.

Abstract

We consider a correlated $N\times N$ Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one.