Outliers and bounded rank perturbation for non-Hermitian random band matrices

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 $\mathbb{E}[X^2]=\rho\mathbf{1}$, 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 $\rho$. 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 $\rho$. 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.