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

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.