Empirical spectral distributions of sparse random graphs

We study the spectrum of a random multigraph with a degree sequence ${\bf D}_n=(D_i)_{i=1}^n$ and average degree $1 \ll \omega_n \ll n$, 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 $\omega_n^{-1} {\bf D}_n$ converges weakly to a limit $\nu$, under mild moment assumptions (e.g., $D_i/\omega_n$ are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to $\nu\boxtimes \sigma_{\rm sc}$, the free multiplicative convolution of $\nu$ 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 Erd\H{o}s-R\'enyi graph with the target ESD, using three intermediate random graphs, with a negligible fraction of edges modified in each step.