On near-critical and dynamical percolation in the tree case

Consider independent bond percolation with retention probability p on a spherically symmetric tree Γ. Write θΓ(p) for the probability that the root is in an infinite open cluster, and define the critical value pc = inf{p : θΓ(p) > 0}. If θΓ(pc) = 0, then the root may still percolate in the corresponding dynamical percolation process at the critical value pc, as demonstrated recently by Häggström, Peres and Steif. Here we relate this phenomenon to the near-critical behaviour of θΓ(p) by showing that the root percolates in the dynamical percolation process if and only if ∫ 1 pc (θΓ(p)) dp < ∞. The “only if” direction extends to general trees, whereas the “if” direction fails in this generality.