Spectral norm of matrices with independent entries up to polyloglog
Rafał Meller
TL;DR
The paper studies the operator norm of the random matrix $(a_{ij} X_{ij})_{i,j\le n}$ with independent symmetric entries under the moment-growth condition $\|X_{ij}\|_{2p} \le \kappa \|X_{ij}\|_p$. The main result gives an upper bound $\mathbb{E}\| (a_{ij} X_{ij}) \|_{op} \lesssim^\kappa \mathrm{Log}^{C(\kappa)}\mathrm{Log}(n)\big(M(A) + \sup_{v,w\in B_2} \|\sum a_{ij} v_i w_j X_{ij}\|_{\mathrm{Log}\, n}\big)$, with a corresponding near two-sided bound, expressed via $M(A)=\max_i \|(a_{ij})_j\|_2 + \max_j \|(a_{ij})_i\|_2$ and a tail-quantified term $R_X(A)$. A key tool is the SRV$(\kappa)$ class together with a graph-degree parameter $d_A$ and a product-decomposition technique that reduces general distributions to subgaussian factors and propagates the bound by induction. The results extend the Latala–Świątkowski program to an intermediate tail regime between subgaussian and heavy-tailed distributions and yield explicit, computable bounds for operator norms of inhomogeneous random 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.