On near-critical and dynamical percolation in the tree case
release_khmyfs7zpja4lfryatu2hnvdvy
by
Olle Haggstrom, Robin Pemantle
2004
Abstract
Consider independent bond percolation with retention probability p on a
spherically symmetric tree Gamma. Write theta_Gamma(p) for the probability that
the root is in an infinite open cluster, and define the critical value
p_c=infp:theta_Gamma(p)>0. If theta_Gamma(p_c)=0, then the root may still
percolate in the corresponding dynamical percolation process at the critical
value p_c, as demonstrated recently by Haggstrom, Peres and Steif. Here we
relate this phenomenon to the near-critical behaviour of theta_Gamma(p) by
showing that the root percolates in the dynamical percolation process if and
only if int_p_c^1 (theta_Gamma(p))^-1dp<infty. The "only if" direction
extends to general trees, whereas the "if" direction fails in this
generality.
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