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We confirm the long-standing prediction that c=e?2.718 is the threshold for the emergence of a non-vanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erd?s-Renyi random graph with average degree c. This is achieved by a detailed second-order analysis of the resolvent (A?z)?1 near the singular point z=0, where A is the adjacency operator of the Poisson-Galton-Watson tree with mean offspring c. More generally, our method applies to arbitrary unimodular Galton-Watson trees, yielding explicit criteria for the presence or absence of extended states at zero in the limiting spectral measure of a variety of random graph models, in terms of the underlying degree distribution. Joint work with Simon Coste.

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• Probabilités, Statistiques mathématiques, Mathématiques appliquées (519)

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