Extreme eigenvalues and eigenvectors for finite rank additive deformations of non-hermitian sparse random matrices

Abstract: Consider a $n\times n$ sparse non-Hermitian random matrix $X_n$ defined as the Hadamard product between a random matrix with centered independent and identically distributed entries and a sparse Bernoulli matrix with success probability $K_n/n$ where $K_n\le n$ (and possibly $K_n\ll n$) and $K_n\to \infty$ as $n\to \infty$. Let $E_n$ be a deterministic $n\times n$ finite-rank matrix. We prove that the outlier eigenvalues of $Y_n= X_n +E_n$ asymptotically match those of $E_n$. In the special case of a rank-one deformation, assuming further that the sparsity parameter satisfies $K_n \gg \log^9(n)$ and that the entries of the random matrix are sub-Gaussian, we describe the limiting behavior of the projection of the right eigenvector associated with the leading eigenvalue onto the right eigenvector of the rank-one deformation. In particular, we prove that the projection behaves as in the Hermitian case. To that end, we rely on the recent universality results of Brailovskaya and van Handel (2024) relating the singular value spectra of deformations of $X_n$ to Gaussian analogues of these matrices. Our analysis builds upon a recent framework introduced by Bordenave et.al. (2022), and amounts to showing the asymptotic equivalence between the reverse characteristic polynomial of the random matrix and a random analytic function on the unit disc with explicit dependence on the finite-rank deformation.