A Graphon Approach to Limiting Spectral Distributions of Wigner-type Matrices

TL;DR

This paper introduces a graphon-based method to determine the limiting spectral distributions of Wigner-type matrices, providing explicit equations and weaker convergence assumptions, with applications to various random graph models.

Contribution

It develops a novel graphon approach for spectral analysis of Wigner-type matrices, improving upon previous methods by relaxing convergence assumptions and applying to diverse sparse graph models.

Findings

Derived explicit moments and Stieltjes transform equations for limiting measures.

Proved the semicircle law for generalized Wigner matrices.

Determined spectral distributions for several sparse inhomogeneous random graph models.

Abstract

We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner-type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes transforms explicitly with weaker assumptions on the convergence of variance profiles than previous results. As applications, we give a new proof of the semicircle law for generalized Wigner matrices and determine the limiting spectral distributions for three sparse inhomogeneous random graph models with sparsity $\omega(1/n)$: inhomogeneous random graphs with roughly equal expected degrees, $W$-random graphs and stochastic block models with a growing number of blocks. Furthermore, we show our theorems can be applied to random Gram matrices with a variance profile for which we can find the limiting spectral distributions under weaker assumptions than previous results.