Delocalization and limiting spectral distribution of $Erd\H{o}s-R\'{e}nyi$ graphs with constant expected degree

Abstract

For fixed $\lambda>0$, it is known that the adjacency matrices of $Erd\H{o}s-R\'{e}nyi$ graphs $\{G(n,\lambda/n),n\in\N\}$, with edge-weights $\lambda^{-1/2}$, have a limiting spectral distribution $\nu_{\lambda}$ as $n\to\infty$. We show ${\nu_{\lambda}}$ converges weakly to the semicircle distribution as $\lambda\to\infty$. Also, when an arbitrarily small positive real $\epsilon>0$ is given, we prove that, for large $\lambda$, there is an orthonormal eigenvector basis of ${G(n,\lambda/n),n\in\N\}$ such that most of the elements in the basis have an infinity norm smaller than $\epsilon$.