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 p T=p T(b,θ)∈(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 ↓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∈(q T,1) converges, as h ↓0, to a well-defined measure