Random matrix statistics appear across diverse fields and are now recognized as exhibiting universal behavior. However, even within random matrix theory itself, the mechanism underlying the universality of local eigenvalue statistics beyond the mean-field setting remains poorly understood.

In this talk, we consider symmetric and Hermitian random matrices whose entries are independent random variables with an arbitrary variance pattern. Under a novel Short-to-Long Mixing condition—sharp in the sense that it excludes any deterministic correction at the spectral edge—we establish GOE/GUE edge universality for such inhomogeneous random matrices, which may be sparse or far from the classical mean-field regime. This condition reduces the universality problem to verifying the mixing properties of Markov chains defined by the variance profile matrix. This talk is based on joint work with Dang-Zheng Liu.