Probability Colloquium Augsburg-Munich
Speakers:
- Amol Aggarwal (Stanford)
- Johannes Alt (Bonn)
Schedule:
14:30 Welcome
15:00 Talk Amol Aggarwal
16:00 Refreshment break
16:30 Talk Johannes Alt
18:30 Option for common dinner
Titles and abstracts:
Amol Aggarwal: Asymptotics for the Maximal Representation of the Symmetric Group
We analyze the behavior of the largest dimension of an irreducible representation of the symmetric group S_N, as N tends to infinity, showing that it behaves as (N!)^{1/2} * e^{-(d+o(1)) N^{1/2}}, for some constant d>0. The proof is based on viewing these dimensions as discrete variants of random matrix integrals or Coulomb gases, and setting up and analyzing a variational principle for their local statistics.
Johannes Alt: Random Graphs: From Spectra to Quantum Dynamics
We study the Erdős-Rényi graph, a common random graph model with intricate behaviours.
First, I establish a transition in the eigenvector structure of its adjacency matrix. For large enough expected degree, all eigenvectors are delocalized, i.e. their mass spreads out approximately uniformly over all vertices. When the expected degree is small enough, localized eigenvectors emerge. More precisely, the mass of each eigenvector with large eigenvalue is concentrated in a ball around a unique vertex and decays exponentially in the graph distance from this vertex.
Second, I determine the Schrödinger time evolution of a free quantum particle hopping on the Erdős-Rényi graph through refined spectral information.
Third, I demonstrate that for the corresponding directed Erdős-Rényi graph, all eigenvectors, with eigenvalue away from zero, are delocalized. This is in contrast to the undirected setting in the first part.
The talk is based on joint works with Raphael Ducatez, Antti Knowles and Sarah Timhadjelt.