Preprint:
Dynamical Percolation on General Trees

D. Khoshnevisan

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, Peres and Steif (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 necessarily spherically symmetric. This answers a question of Yuval Peres (personal communication). We present also calculations of the Hausdorff dimension of exceptional times of percolation.

Keywords. Dynamical percolation; capacity; trees.

AMS Classification (2000) Primary. 60K35; Secondary. 31C15, 60J45.

Support. Research supported in part by a grant from the National Science Foundation.

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Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.
davar@math.utah.edu

Updates: January 17, 2007; May 25, 2006
© 2006 - Davar Khoshnevisan