Edge Universality for Inhomogeneous Random Matrices

TL;DR

This paper proves edge universality for a broad class of inhomogeneous random matrices under a novel mixing condition, extending classical results to highly inhomogeneous, sparse, and non-mean-field ensembles.

Contribution

It introduces a sharp Short-to-Long Mixing condition that reduces universality verification to mixing properties of a variance profile-based random walk.

Findings

Establishes GOE/GUE edge universality for inhomogeneous matrices.

Applies to sparse, inhomogeneous, and heavy-tailed ensembles.

Broadens the scope of universality results beyond classical settings.

Abstract

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 $d\geq N^{2/3+\epsilon}$ for any $\epsilon>0$.