Abstract
Häggström et al. (Ann Inst H Poincaré Probab Stat 33(4):497–528, 1997) have introduced a dynamical version of percolation on a graph G. When G is a tree they derived a necessary and sufficient condition for percolation to exist at some time t. In the case that G is a spherically symmetric tree (Peres and Steif in Probab Theory Relat Fields 111(1):141–165, 1998), derived a necessary and sufficient condition for percolation to exist at some time t in a given target set D. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time \({t\in D}\), in the case that the underlying tree is not necessary spherically symmetric. This answers a question of Yuval Peres (personal communication). We present also a formula for the Hausdorff dimension of the set of exceptional times of percolation.
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Research supported in part by a grant from the National Science Foundation.
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Khoshnevisan, D. Dynamical percolation on general trees. Probab. Theory Relat. Fields 140, 169–193 (2008). https://doi.org/10.1007/s00440-007-0061-6
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DOI: https://doi.org/10.1007/s00440-007-0061-6
Keywords
- Dynamical percolation
- Capacity
- Trees
Mathematics Subject Classification (2000)
- Primary: 60K35
- Secondary: 31C15
- Secondary: 60J45