Bikker R.-P.
Semi-oriented bootstrap percolation in three dimensions
(187K, Postscript)
ABSTRACT. We consider the critical system size of a three dimensional semi-oriented
bootstrap percolation model, constricted to a 3$D$ cube wrapped to a
torus, i.e.\ with periodical boundary conditions. We point out a possible
form of the critical droplets for this model: occupied squares in a plane
perpendicular to the primary direction of the dynamics behave as growing
seeds when they are sufficiently large. Their critical linear size is of
order $\exp O({1 \over p} \log^2 {1 \over p})$. Also we prove that
$\exp \exp O({1 \over p} \log^2 {1 \over p})$ is an upper bound on the
critical system size. In the proof an element comes forward that is not
contained in the two dimensional SBP model. Whereas in two dimensions a
critical droplet can only either grow, survive or die, in three
dimensions a growing critical droplet can also shrink. It follows from
this result that for the SBP$^{2,1}$ model $p_c=0$, i.e.\ on the infinite
lattice this model almost surely fills the whole space for all nonzero $p$.