# Delocalization transition for critical Erdős-Rényi graphs

Johannes Alt, Raphael Ducatez, Antti Knowles

Johannes Alt, Raphael Ducatez, Antti Knowles

**28/5/20**Published in : arXiv:2005.14180

We analyse the eigenvectors of the adjacency matrix of a critical Erdős-Rényi graph

\mathbb G(N,d/N) |

, where *d *is of order* log N*. We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent

\gamma(\mathbf w) |

of an eigenvector *w*, defined through

\|\mathbf w\|_\infty / \|\mathbf w\|_2 = N^{-\gamma(\mathbf w)} |

. Our results remain valid throughout the optimal regime

\sqrt{\log N} \ll d \leq O(\log N) |