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.
Pre/E-Prints. This paper is available in
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