Metastable Behavior for Bootstrap Percolation on Regular Trees

We examine bootstrap percolation on a regular (b + 1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least θ occupied neighbors, occupied sites remain occupied forever. It is known that, when b > θ ≥ 2, the limiting density q = q(p) of occupied sites exhibits a jump at some pT = pT(b, θ) ∈ (0, 1) from qT := q(pT) < 1 to q(p) = 1 when p > pT. We investigate the metastable behavior associated with this transition. Explicitly, we pick p = pT + h with h > 0 and show that, as h ↓ 0, the system lingers around the “critical” state for time order h−1/2 and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q ∈ (qT, 1) converges, as h ↓ 0, to a well-defined measure.