Empirical Spectral Distributions of Sparse Random Graphs

We study the spectrum of a random multigraph with a degree sequence Dn=(Di)i=1n and average degree 1 ≪ ω_n ≪ n, generated by the configuration model, and also the spectrum of the analogous random simple graph. We show that, when the empirical spectral distribution (ESD) of ωn−1Dn converges weakly to a limit ν, under mild moment assumptions (e.g., D_i∕ω_n are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to ν⊠ σ_SC, the free multiplicative convolution of ν with the semicircle law. Relating this limit with a variant of the Marchenko–Pastur law yields the continuity of its density (away from zero), and an effective procedure for determining its support. Our proof of convergence is based on a coupling between the random simple graph and multigraph with the same degrees, which might be of independent interest. We further construct and rely on a coupling of the multigraph to an inhomogeneous Erdős-Rényi graph with the target ESD, using three intermediate random graphs, with a negligible fraction of edges modified in each step.