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On near-critical and dynamical percolation in the tree case

Olle Haggstrom·Robin Pemantle
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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=inf{p: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))^{-1}dp<infty. The ``only if'' direction extends to general trees, whereas the ``if'' direction fails in this generality.

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Olle Haggstrom, Robin Pemantle. 2004-04-05. On near-critical and dynamical percolation in the tree case. https://arxiv.org/abs/math/0404091

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