Edge Universality for Inhomogeneous Random Matrices

We consider symmetric and Hermitian random matrices whose entries are independent and symmetric random variables with an arbitrary variance pattern. Under a novel Short-to-Long Mixing condition, which is sharp in the sense that it precludes a corrected shift at the spectral edge, we establish GOE/GUE edge universality for such inhomogeneous random matrices. This condition effectively reduces the universality problem to verifying the mixing properties of a random walk governed by the variance profile matrix. Our universality results are applicable to a remarkably broad class of random matrix ensembles that may be highly inhomogeneous, sparse, or far beyond the mean-field setting of classical random matrix theory. Notable examples include: 1. Inhomogeneous Wishart-type random matrices; 2. Random band matrices whose entries are independent random variables with general variance profile, particularly with an optimal bandwidth in dimensions $d \le 2$; 3. Sparse random matrices with structured variance profiles; 4. Generalized Wigner matrices under significantly weaker sparsity constraints and heavy-tailed entry distributions; 5. Wegner orbital models under sharp mixing assumptions; 6. Random 2-lifts of random $d$-regular graphs where $d\geq N^{2/3+ε}$ for any $ε>0$.