# Vortrag

Modul:   MAT971  Stochastische Prozesse

## Spectral phases of Erdös-Rényi graphs

Vortrag von Prof. Dr. Antti Knowles

Datum: 02.11.22  Zeit: 17.15 - 18.15  Raum: ETH HG G 19.1

Disordered quantum systems exhibit a variety of spectral phases, characterized by the extent of spatial localization of the eigenvectors. Through their adjacency matrices, random graphs provide a natural class of models for such systems, where the disorder arises from the random geometry of the graph. The simplest random graph is the Erdös-Rényi graph G(N,p), whose adjacency matrix is the archetypal sparse random matrix. The parameter d=pN represents the expected degree of a vertex. A dramatic change in behaviour is known to occur at the scale d \sim \log N, which is the threshold where the degrees of the vertices cease to concentrate. Below this scale the graph becomes inhomogeneous and develops structures such as hubs and leaves which accompany the appearance of a localized phase.

I report on recent progress in establishing the phase diagram for G(N,p) at and below the critical scale d \sim \log N. We show that the spectrum splits into a fully delocalized region in the middle of the spectrum and a semilocalized phase near the spectral edges. The transition between the phases is sharp in the sense of a discontinuity in the localization exponent of eigenvectors. Furthermore, we show that the semilocalized phase consists of a fully localized region and in addition, for some values of d, a complementary region that we conjecture to be nonergodic delocalized.

Joint work with Johannes Alt and Raphael Ducatez.