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In dynamical critical site percolation on the triangular lattice or bondpercolation on \Z^2, we define and study a local time measure on theexceptional times at which the origin is in an infinite cluster. We show thatat a typical time with respect to this measure, the percolation configurationhas the law of Kesten's Incipient Infinite Cluster. In the most technicalresult of this paper, we show that, on the other hand, at the first exceptionaltime, the law of the configuration is different. We also study the collapse ofthe infinite cluster near typical exceptional times, and establish a relationbetween static and dynamic exponents, analogous to Kesten's near-criticalrelation.

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