A two-dimensional binary (
) totalistic
cellular automaton with a von Neumann
neighborhood of range 
. It has a birth rule that at least 2 of its 4 neighbors
are alive, and a survival rule that all cells survive. 
steps of bootstrap percolation on an 
grid with random initial condition of density 
can be implemented in the Wolfram
Language as
With[{n = 10, p = 0.1, s = 20},
CellularAutomaton[
{1018, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}},
{1, 1}},
Table[If[Random[Real] < p, 1, 0], {s}, {s}],
n
]
]
If the initial condition consists of a random sparse arrangement of cells with density 
,
then the system seems to quickly converge to a steady state of rectangular islands
of live cells surrounded by a sea of dead cells. However, as 
crosses some threshold on finite-sized grids, the behavior
appears to change so that every cell becomes live. Several examples are shown above
on three 
grids with random initial conditions and different starting densities.
However, this conclusion proves to be incorrect, since the apparent change in behavior actually is a spurious edge effect introduced by use of a finite-sized grid. Amazingly, Holroyd (2003) showed that the asymptotic threshold occurs such that
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