Events Archive

15:15: Johannes Alt, Uni Bonn

Spectral Phases of the Erdős–Rényi graph

Abstract: We consider the Erdős–Rényi graph on N vertices with expected degree d for each vertex. It is well known that the structure of this graph changes drastically when d is of order log N. Below this threshold it develops inhomogeneities which lead to the emergence of localized eigenvectors, while the majority of eigenvectors remains delocalized. In this talk, I will explain our results in both phases and present the phase diagram depicting them. For a certain regime in d, we establish a mobility edge by showing that the localized phase extends up to the boundary of the delocalized phase. This is based on joint works with Raphael Ducatez and Antti Knowles.

16:45: Torben Krüger, FAU

Merging singularities in two-dimensional Coulomb gases

The two-dimensional one-component plasma is a particle system in the plane with long-range logarithmic interactions. At a specific temperature the system is equivalent to the eigenvalue ensemble of a normal random matrix model. In equilibrium the particles form distinct droplets when placed in an external potential. Using the Riemann-Hilbert approach we determine the local statistical behaviour of the particles at the point where two droplets merge and observe an anisotropic scaling behaviour with particles being much further apart in the direction of merging than the perpendicular direction. This observation lends support to the conjecture that the hierarchy of local particle statistics at singularities of the density of states within two-dimensional Coulomb gases coincides with the corresponding hierarchy of one-dimensional invariant ensembles. This is joint work with Meng Yang and Seung-Yeop Lee