Delocalization and Limiting Spectral Distribution of Erd\H{o}s-R\'{e}nyi Graphs with Constant Expected Degree

We consider Erd\H{o}s-R'{e}nyi graphs $G(n,p_n)$ with large constant expected degree $\lambda$ and $p_n=\lambda \mathrm{/}n$. Bordenave and Lelarge (2010) showed that the infinite-volume limit, in the Benjamini-Schramm topology, is a Galton-Watson tree with offspring distribution Pois($\lambda$) and the mean spectrum at the root of this tree has unbounded support and corresponds to the limiting spectral distribution of $G(n,p_n)$ as $n\to \mathrm{\infty }$. We show that if one weights the edges by $1\mathrm{/}\sqrt{\lambda }$ and sends $\lambda \to \mathrm{\infty }$, then the support mostly vanishes and in fact, the limiting spectral distributions converge weakly to a semicircle distribution. We also find that for large $\lambda$, there is an orthonormal eigenvector basis of $G(n,p_n)$ such that most of the vectors delocalize with respect to the infinity norm, as $n\to \mathrm{\infty }$. Our delocalization result provides a variant on a result of Tran, Vu and Wang (2013).