Outliers and bounded rank perturbation for non-Hermitian random band matrices
Yi Han
TL;DR
The paper studies spectral properties of general non-Hermitian random matrices with inhomogeneous variance profiles, including band matrices, and proves that eigenvalues largely lie near the elliptic law support $ ext{E}_ ho$ while a finite-rank perturbation creates a finite number of outliers. A master theorem, proven first for Gaussian matrices and extended to non-Gaussian ones via universality, combines matrix Dyson equations, free probability, and isotropic resolvent laws to control the spectrum and to locate outliers via a determinant criterion. The results extend prior outlier theorems from i.i.d. and elliptic models to highly sparse, inhomogeneous, and banded matrices, and further apply to products of independent elliptic matrices, with explicit convergence rates and robust truncation arguments. Overall, the work provides a unified framework for understanding bulk spectral constraints and finite-rank perturbations in a broad class of non-Hermitian random matrices with applications to band-structured graphs and matrix products.
Abstract
In this work we consider general non-Hermitian square random matrices $X$ that include a wide class of random band matrices with independent entries. Whereas the existence of limiting density is largely unknown for these inhomogeneous models, we show that spectral outliers can be determined under very general conditions when perturbed by a finite rank deterministic matrix. More precisely, we show that whenever $\mathbb{E}[X]=0,\mathbb{E}[XX^*]=\mathbb{E}[X^*X]=\mathbf{1}$ and $\mathbb{E}[X^2]=ρ\mathbf{1}$, and under mild conditions on sparsity and entry moments of $X$, then with high possibility all eigenvalues of $X$ are confined in a neighborhood of the support of the elliptic law with parameter $ρ$. Also, a finite rank perturbation property holds: when $X$ is perturbed by another deterministic matrix $C_N$ with bounded rank, then the perturbation induces outlying eigenvalues whose limit depends only on outlying eigenvalues of $C_N$ and $ρ$. This extends the result of Tao on i.i.d. random matrices and O'rourke and Renfrew on elliptic matrices to a family of highly sparse and inhomogeneous random matrices, including all Gaussian band matrices on regular graphs with degree at least $(\log N)^3$. A quantitative convergence rate is also derived. We also consider a class of finite rank deformations of products of at least two independent elliptic random matrices, and show it behaves just as product i.i.d. matrices.