Abstract
Let a low densityp of sites on the lattice Z2 be occupied, remove a proportionq of them, and call the remaining sites empty. Then update this configuration in discrete time by iteration of the following synchronous rule: an empty site becomes occupied by contact with at least two occupied nearest neighbors, while occupied and removed sites nerver change their states. Ifq/p 2 is large most sites remain unoccupied forever, while ifq/p 2 is small, this dynamics eventually makes most sites occupied. This demonstrates how sensitive the usual bootstrap percolation rule (theq=0 case) is to the pollution of space.
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Gravner, J., McDonald, E. Bootstrap percolation in a polluted environment. J Stat Phys 87, 915–927 (1997). https://doi.org/10.1007/BF02181252
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DOI: https://doi.org/10.1007/BF02181252