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 theta occupied neighbors, occupied sites remain occupied forever. It is known that, when b >theta a parts per thousand yen2, the limiting density q=q(p) of occupied sites exhibits a jump at some p (T)=p (T)(b,theta)a(0,1) from q (T):=q(p (T))< 1 to q(p)=1 when p > p (T). We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p (T)+h with h > 0 and show that, as h a dagger"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 qa(q (T),1) converges, as h a dagger"0, to a well-defined measure.