THE NUMBER OF INFINITE CLUSTERS IN DYNAMICAL PERCOLATION

Dynamical percolation is a Markov process on the space of subgraphs of a given graph, that has the usual percolation measure as its stationa ry distribution. In previous work with O. Haggstrom, we found conditio ns for existence of infinite clusters at exceptional times. Here we sh ow that for Z(d), with p > p(c), a.s. simultaneously for all times the re is a unique infinite cluster, and the density of this cluster is th eta(p). For dynamical percolation on a general tree Gamma, we show tha t for p > p(c), a.s. there are infinitely many infinite clusters at al l times. At the critical value p = p(c), the number of infinite cluste rs may vary, and exhibits surprisingly rich behaviour. For spherically symmetric trees, we find the Hausdorff dimension of the set T-k Of ti mes where the number of infinite clusters is k, and obtain sharp capac ity criteria for a given time set to intersect T-k. The proof of this capacity criterion is based on a new kernel truncation technique.