Alexander Drewitz: (Near-)critical behavior of a strongly correlated percolation model

Abstract: For (near-)critical independent Bernoulli percolation, particularly profound results have been obtained in the high-dimensional setting as well as on planar lattices. We consider a strongly correlated percolation model — the level sets of the metric graph Gaussian free field — where significant understanding has also been developed regarding its (near-)critical behavior in intermediate dimensions. We will explain the origin of the model's integrability, and discuss its implications for the associated universality class.

Benedikt Jahnel: First contact percolation

The talk presents a version of first passage percolation on $\Z^d$ where the random passage times on the edges are replaced by contact times given by random closed sets on $\R$. Similarly to the contact process without recovery, an infection can spread into the system along increasing sequences of contact times. In case of stationary contact times, we can identify associated first passage percolation models, which in turn establish shape theorems also for first contact percolation. In case of periodic contact times that reflect some reoccurring daily pattern, we also present shape theorems with limiting shapes that are universal with respect to the within-one-day contact distribution. In this case, we also prove a Poisson approximation for increasing numbers of within-one-day contacts. Finally, we present a comparison of the limiting speeds of three models - all calibrated to have one expected contact per day - that suggests that less randomness is beneficial for the speed of the infection. The proofs rest on coupling and subergodicity arguments. This is joint work with Anh Duc Vu and Lukas Lüchtrath.