Edge Universality for Inhomogeneous Random Matrices II: Markov Chain Comparison and Critical Statistics

TL;DR

This paper extends the analysis of edge universality in inhomogeneous random matrices by developing Markov chain comparison techniques applicable across different sparsity regimes, revealing universal and non-universal spectral phenomena.

Contribution

It introduces new comparison conditions that relate spectral edge statistics of matrices via their Markov variance profiles, broadening universality results to subcritical and critical regimes.

Findings

Edge statistics are universal when Markov chains are comparable.

Comparison theorems apply to models like band matrices and the Wegner orbital model.

Non-universal phenomena depend on geometric and spike structures.

Abstract

The first paper in this series introduced a \emph{short-to-long mixing} condition that captures mean-field GOE/GUE edge universality in the supercritical sparsity regime, for symmetric/Hermitian random matrices with independent entries and a Markov variance profile. This condition reduces the universality problem to the mixing properties of the underlying Markov chains. In this paper, we develop new \emph{short-to-long comparison} conditions that extend the analysis to the subcritical and critical sparsity regimes. Specifically, we prove that two inhomogeneous random matrices exhibit the same universal edge statistics whenever their variance-profile Markov chains are comparable, regardless of the fine details of the matrix entries. To illustrate the power of our Markov chain comparison theorem, we derive the spectral edge statistics for several prototypical models: random band matrices, the Wegner orbital model, and Hankel-profile random matrices. These comparisons uncover a rich landscape of both universal and non-universal phenomena -- shaped by geometric structure, spike patterns, and domains of stable attraction -- features that lie fundamentally beyond the reach of classical random matrix theory.