A semicircle law for the normalized Laplacian of sparse random graphs

Abstract

We study the limiting spectral distribution of the normalized Laplacian $\mathcal L$ of an Erd\H{o}s-R\'enyi graph $G(n,p)$. To account for the presence of isolated vertices in the sparse regime, we define $\mathcal L$ using the Moore-Penrose pseudoinverse of the degree matrix. Under this convention, we show that the empirical spectral distribution of a suitably normalized $\mathcal L$ converges weakly in probability to the semicircle law whenever $np\to\infty$, thereby providing a rigorous justification of a prediction made in (Akara-pipattana and Evnin, 2023). Moreover, if $np>\log n+\omega(1)$, so that $G(n,p)$ has no isolated vertices with high probability, the same conclusion holds for the standard definition of $\mathcal L$. We further strengthen this result to almost sure convergence when $np=\Omega(\log n)$. Finally, we extend our approach to the Chung-Lu random graph model, where we establish a semicircle law for $\mathcal L$ itself, improving upon (Chung, Lu, and Vu 2003), which obtained the semicircle law only for a proxy matrix.