Local time on the exceptional set of dynamical percolation, and the Incipient Infinite Cluster

Understand

In dynamical critical site percolation on the triangular lattice or bond percolation on \Z^2, we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster.

• We show that at a typical time with respect to this measure, the percolation configuration has the law of Kesten's Incipient Infinite Cluster.
• In the most technical result of this paper, we show that, on the other hand, at the first exceptional time, the law of the configuration is different.
• We also study the collapse of the infinite cluster near typical exceptional times, and establish a relation between static and dynamic exponents, analogous to Kesten's near-critical relation.