On near-critical and dynamical percolation in the tree case release_khmyfs7zpja4lfryatu2hnvdvy

by Olle Haggstrom, Robin Pemantle

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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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Date   2004-04-05
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arXiv  math/0404091v1
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