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arXiv · 0705.0140

Dynamical percolation on general trees

Davar Khoshnevisan
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Abstract

Häggström, Peres, and Steif (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, Häggström, Peres, and Steif (1997) 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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Davar Khoshnevisan. 2007-05-01. Dynamical percolation on general trees. https://doi.org/10.1007/s00440-007-0061-6

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