Pete Gábor (BME TTK Sztochasztika Tanszék): "Local time on the exceptional set of dynamical percolation, and the Incipient Infinite Cluster"

In critical planar percolation, there are almost surely no infinite clusters. However, if the configuration evolves according to a continuous time Markov chain, there could be random exceptional times when the origin is connected to infinity. A theorem of Christophe Garban, Oded Schramm and myself from 2008 is that such exceptional times do exist, and (for site percolation on the triangular lattice) their Hausdorff dimension is 31/36.

How does the cluster of the origin look like at exceptional times? In joint work with Alan Hammond and Oded Schramm, we define a notion of a typical exceptional time, and we show that, at such a time, the law of the infinite cluster is Kesten's Incipient Infinite Cluster. On the other hand, the cluster of the origin at the very first exceptional time looks different.