Outliers for deformed inhomogeneous random matrices

Abstract

Inhomogeneous random matrices with non-trivial variance profiles determined by symmetric stochastic matrices and with independent sub-Gaussian entries up to Hermitian symmetry, encompass a wide range of important models, including sparse Wigner matrices and random band matrices. In these models, the maximum entry variance-a natural proxy for sparsity-serves both as a key structural feature and a primary analytical obstacle. In this paper, we consider low-rank additive perturbations of such matrices and establish a sharp BBP phase transition for extreme eigenvalues at the level of the law of large numbers. Furthermore, in the Gaussian setting, we derive the fluctuations of spectral outliers under suitable conditions on the variance profile and perturbation. These fluctuations exhibit strong non-universality, depending on the eigenvectors, sparsity levels, and the underlying geometric structure. Our proof strategies rely on ribbon graph expansions, upper bounds for diagram functions, large-moment estimates, and the enumeration of typical diagrams.