Limiting empirical spectral measure of the normalized Laplacian in preferential attachment graphs

TL;DR

This paper proves that the empirical spectral distribution of the normalized Laplacian in preferential attachment graphs converges to a deterministic measure, characterized via the local weak limit and Green function analysis.

Contribution

It introduces a novel approach combining resolvent methods, local weak limits, and concentration techniques to analyze spectral measures in preferential attachment graphs.

Findings

Spectral distribution converges to a deterministic measure on [0, 2]

Limit characterized by the Green function of the Pólya-point graph

Methodology combines resolvent analysis with probabilistic concentration

Abstract

We study the empirical spectral distribution of the normalized Laplacian of linear preferential attachment graphs in the Barab{\'a}si-Albert regime with fixed out-degree. For the resulting sequence of random multigraphs, we prove that the empirical spectral distribution converges weakly in probability to a deterministic probability measure supported on the interval [0, 2]. The limit is characterized via the local weak limit of preferential attachment graphs (the P{\'o}lya-point graph): the limiting Stieltjes transform is given by the expected diagonal Green function at the root of the normalized Laplacian operator on this infinite random graph. The proof combines a resolvent approach with a uniform Neumann-series expansion for the normalized Laplacian, a random-walk representation in terms of return probabilities on decorated neighborhoods, a truncation and Doob martingale-Azuma-Hoeffding concentration argument along the PA filtration, and an analytic continuation argument based on normal families.