Delocalization transition for critical Erdös-Rényi graphs

27 October 2020
15:30
Antti Knowles

Further Information: 

Further Information: 

This seminar will be held via zoom. Meeting link will be sent to members of our mailing list (https://lists.maths.ox.ac.uk/mailman/listinfo/random-matrix-theory-announce) in our weekly announcement on Monday.

Abstract

We analyse the eigenvectors of the adjacency matrix of a critical Erdös-Rényi graph 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 exponents of the eigenvectors. Joint work with Johannes Alt and Raphael Ducatez.

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