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Delocalization and Limiting Spectral Distribution of Erdős-Rényi Graphs with Constant Expected Degree

We consider Erdős-Rényi graphs $G(n,p_n)$ with large constant expected degree $λ$ and $p_n=λ/n$. Bordenave and Lelarge (2010) showed that the infinite-volume limit, in the Benjamini-Schramm topology, is a Galton-Watson tree with offspring distribution Pois($λ$) 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 $n\to\infty$. We show that if one weights the edges by $1/\sqrtλ$ and sends $λ\to\infty$, then the support mostly vanishes and in fact, the limiting spectral distributions converge weakly to a semicircle distribution. We also find that for large $λ$, 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 $n\to\infty$. Our delocalization result provides a variant on a result of Tran, Vu and Wang (2013).