On near-critical and dynamical percolation in the tree case

Consider independent bond percolation with retention probability p on a sph erically symmetric tree Gamma. Write theta(Gamma)(P) for the probability th at 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 roo t 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 behavior of the ta(Gamma)(p) by showing that the root percolates in the dynamical percolati on process if and only if integral(pc)(1)(theta(Gamma)(p))(-1) dp < infinit y. The "only if" direction extends to general trees, whereas the "if" direc tion fails ire this generality. (C) 1999 John Wiley & Sons, Inc.