Linear Lower Bounds for delta(c)(p) for a Class of 2D Self-Destructive Percolation Models
The self-destructive percolation model is defined as follows: Consider percolation with parameter p > p(c) Remove the infinite occupied cluster. Finally, give each vertex (or, for bond percolation, each edge) that at this stage is vacant, an extra chance delta to become occupied. Let delta(c)(p) be the minimal value of delta, needed to obtain an infinite occupied cluster in the final configuration. This model was introduced by van den Berg and Brouwer. They showed, for the site model on the square lattice (and a few other 2D lattices satisfying a special technical condition) that delta(c)(p) >= (p-p(c))/p. In particular, delta(c)(p) is at least linear in p - p(c). Although the arguments used by van den Berg and Brouwer look very lattice-specific, we show that they can be suitably modified to obtain similar linear lower bounds for delta(c)(p) (with p near p(c)) for a much larger class of 2D lattices, including bond percolation on the square and triangular lattices, and site percolation on the star lattice (or matching lattice) of the square lattice. (C) 2009 Wiley Periodicals, Inc. Random Strict. Alg., 34, 520-526, 2009
self-destructive percolation percolation critical value msc numbers: 60k35, 82843