  We study a variant of bootstrap percolation in which growth is restricted to
a single active cluster. Initially there is a single active site at the origin,
while other sites of Z^2 are independently occupied with small probability p,
otherwise empty. Subsequently, an empty site becomes active by contact with 2
or more active neighbors, and an occupied site becomes active if it has an
active site within distance 2. We prove that the entire lattice becomes active
with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 + c sqrt p
and pi^2/9 + C sqrt p (-log p)^3. This corrects previous numerical predictions
for the scaling of the correction term.
