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.