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Dynamical percolation on general trees

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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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Correspondence to Davar Khoshnevisan.

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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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  • Dynamical percolation
  • Capacity
  • Trees

Mathematics Subject Classification (2000)

  • Primary: 60K35
  • Secondary: 31C15
  • Secondary: 60J45