Bootstrap percolation on homogeneous trees has 2 phase transitions

We study the threshold θ bootstrap percolation model on the homogeneous tree with degree b+ 1, 2 ≤ θ ≤ b, and initial density p. It is known that there exists a nontrivial critical value for p, which we call pf , such that a) for p > pf , the final bootstrapped configuration is fully occupied for almost every initial configuration, and b) if p < pf , then for almost every initial configuration, the final bootstrapped configuration has density of occupied vertices less than 1. In this paper, we establish the existence of a distinct critical value for p, pc, such that 0 < pc < pf , with the following properties: 1) if p ≤ pc, then for almost every initial configuration there is no infinite cluster of occupied vertices in the final bootstrapped configuration; 2) if p > pc, then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration. Moreover, we show that 3) for p < pc, the distribution of the occupied cluster size in the final bootstrapped configuration has an exponential tail; 4) at p = pc, the expected occupied cluster size in the final bootstrapped configuration is infinite; 5) the probability of percolation of occupied vertices in the final bootstrapped configuration is continuous on [0, pf ] and analytic on (pc, pf ), admitting an analytic continuation from the right at pc and, only in the case θ = b, also from the left at pf .