BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Antti Knowles (University of Geneva) DTSTART:20210409T133000Z DTEND:20210409T143000Z DTSTAMP:20260423T022715Z UID:MEGA/23 DESCRIPTION:Title: Th e spectral edge of (sub-)critical Erdös-Rényi graphs.\nby Antti Know les (University of Geneva) as part of Séminaire MEGA\n\n\nAbstract\nIt is well known that the Erdős-Rényi graph on N vertices with edge probabili ty d/N undergoes a dramatic change in behaviour when the mean degree d cro sses the critical scale log(N): the degrees of the graph cease to concentr ate about their means and the graph loses its homogeneity. We analyse the eigenvalues and eigenvectors of its adjacency matrix in the regime where t he mean degree d is comparable to or less than the critical scale log(N). We show that the eigenvalue process near the spectral edges is asymptotica lly Poisson\, and the intensity measure is determined by the fluctuations of the large degrees as well as the size of the 2-spheres around vertices of large degree. We conclude that in general the laws of the largest eigen values are not described by the classical Fisher–Tippett–Gnedenko theo rem. As an application of our result\, we prove that the associated eigenv ectors are are exponentially localized in unique\, disjoint balls. Togethe r with the previously established complete delocalization of the eigenvect ors in the middle of the spectrum\, this establishes the coexistence of a delocalized and a localized phase in the critical Erdös-Rényi graph. Joi nt work with Johannes Alt and Raphael Ducatez.\n LOCATION:https://researchseminars.org/talk/MEGA/23/ END:VEVENT END:VCALENDAR