Bounds on the norm of Wigner-type random matrices

TL;DR

This paper derives improved bounds on the spectral norm of Wigner-type random matrices by analyzing the variance structure and employing a Markov chain approximation, advancing understanding of their spectral properties.

Contribution

It provides a significantly sharper bound on the matrix norm in terms of the variance matrix powers, improving previous estimates.

Findings

Established a new bound on the spectral norm involving powers of the variance matrix S.

Introduced an effective Markov chain approximation for analyzing contributions from weighted Dyck paths.

Enhanced theoretical understanding of the spectral behavior of Wigner-type matrices.

Abstract

We consider a Wigner-type ensemble, i.e. large hermitian $N\times N$ random matrices $H=H^*$ with centered independent entries and with a general matrix of variances $S_{xy}=\mathbb E|H_{xy}|^2$. The norm of $H$ is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of $S$ that substantially improves the earlier bound $2\| S\|^{1/2}_\infty$ given in [arXiv:1506.05098]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.