Abstract
Dynamical percolation is a Markov process on the space of subgraphs of a given graph, that has the usual percolation measure as its stationary distribution. In previous work with O. Haggstrom, we found conditions for existence of infinite clusters at exceptional times. Here we show that for ℤ d , with p>p c , a.s. simultaneously for all times there is a unique infinite cluster, and the density of this cluster is θ(p). For dynamical percolation on a general tree Γ, we show that for p>p c , a.s. there are infinitely many infinite clusters at all times. At the critical value p=p c , the number of infinite clusters may vary, and exhibits surprisingly rich behaviour. For spherically symmetric trees, we find the Hausdorff dimension of the set T k of times where the number of infinite clusters is k, and obtain sharp capacity criteria for a given time set to intersect T k . The proof of this capacity criterion is based on a new kernel truncation technique.
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
