97-67 Bikker R.-P.

Semi-oriented bootstrap percolation in three dimensions (187K, Postscript) Feb 12, 97

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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 $p$.

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