Edge Universality for Inhomogeneous Random Matrices

TL;DR

This work establishes sharp edge universality for a broad class of inhomogeneous random matrices by introducing a Short-to-Long Mixing condition that ties spectral-edge statistics to a variance-profile Markov chain. The authors develop a novel Chebyshev polynomial moment expansion and a diagrammatic ribbon-graph framework for Gaussian IRMs, then extend to sub-Gaussian and Wishart-type ensembles, proving Tracy-Widom fluctuations and BBP-type deformations under precise mixing and moment constraints. The approach unifies edge statistics across highly inhomogeneous, sparse, and non-mean-field models, including inhomogeneous Wigner, band, sparse, and block matrices, and random lifts, providing sharp thresholds for universality. Practically, this yields a versatile toolkit to predict extreme eigenvalue behavior in complex networks and structured random systems, with broad implications for physics, statistics, and data science where inhomogeneity is intrinsic.

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 $d\geq N^{2/3+ε}$ for any $ε>0$.

Figures (18)

• Figure 1: Example of one possible gluing in Hermitian case of $\mathbb{E}[\mathrm{Tr} X^{6}\mathrm{Tr} X^{4}\mathrm{Tr} X^{8}]$. The green vertices are marked vertices.
• Figure 2: Example of one possible gluing of $\mathbb{E}[\mathrm{Tr} X^{6}\mathrm{Tr} X^{4}]$ with open edges.
• Figure 3: Okounkov contraction. The green vertex is the marked vertex, and dashed lines represent the Catalan trees. The red line indicates the open edge. Each interior point is traversed twice—hence two dashed lines—since a Catalan tree may grow on each traversal. In the first arrow (steps (i) and (ii) of Definition \ref{['def:Okounkov_contraction']}), all Catalan trees are collapsed and the marked vertex is moved to the tree root. In the second arrow (step (iii)), all degree-2 vertices are removed, yielding the reduced ribbon graph.
• Figure 4: Examples of reduced ribbon graphs (diagrams). In the left lower graph, the circuit starts at the marked vertex and traverses with order $1,2,3$.
• Figure 5: The underlying graphs corresponding to diagrams in Figure \ref{['fig:reduced_ribbon_example']}.

Theorems & Definitions (96)

• Definition 1.1: Inhomogeneous (deformed) random matrices
• Definition 1.2: Short-to-Long Mixing
• Theorem 1.3: Edge Universality
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Definition 2.1: Ribbon graph
• Definition 2.2: Diagram/Reduced ribbon graph
• Definition 2.3: Okounkov contraction
• Proposition 2.4: geng2024outliers