Spectral radius concentration for inhomogeneous random matrices with independent entries

TL;DR

This paper establishes bounds and deviation inequalities for the spectral radius of inhomogeneous random matrices with independent entries, extending understanding beyond i.i.d. cases to more general variance profiles.

Contribution

It provides new upper bounds and deviation inequalities for the spectral radius of inhomogeneous random matrices with independent entries, including heavy-tailed and non-Hermitian cases.

Findings

Spectral radius bounded by 1+ε under optimal sparsity conditions.

Deviation inequalities for spectral radius beyond typical fluctuation scales.

Bounded spectral radius in heavy-tailed regimes with finite moments.

Abstract

Let $A$ be a square random matrix of size $n$, with mean zero, independent but not identically distributed entries, with variance profile $S$. When entries are i.i.d. with unit variance, the spectral radius of $n^{-1/2}A$ converges to $1$ whereas the operator norm converges to 2. Motivated by recent interest in inhomogeneous random matrices, in particular non-Hermitian random band matrices, we formulate general upper bounds for $\rho(A)$, the spectral radius of $A$, in terms of the variance $S$. We prove (1) after suitable normalization $\rho(A)$ is bounded by $1+\epsilon$ up to the optimal sparsity $\sigma_*\gg (\log n)^{-1/2}$ where $\sigma_*$ is the largest standard deviation of an individual entry; (2) a small deviation inequality for $\rho(A)$ capturing fluctuation beyond the optimal scale $\sigma_*^{-1}$; (3) a large deviation inequality for $\rho(A)$ with Gaussian entries and doubly stochastic variance; and (4) boundedness of $\rho(A)$ in certain heavy-tailed regimes with only $2+\epsilon$ finite moments and inhomogeneous variance profile $S$. The proof relies heavily on the trace moment method.