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Linear Lower Bounds for $delta_c(p)$ for a Class of 2D Self-Destructive Percolation Models | | J. van den Berg
; B.N.B. de Lima
; | | Date: |
22 Nov 2007 | | Abstract: | 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 some years ago by van den
Berg and Brouwer. They showed that, for the site model on the square lattice
(and a few other 2D lattices satisfying a special technical condition) that
$delta_c(p)geqfrac{(p-p_c)}{p}$. In particular, $delta_c(p)$ is at least
linear in $p-p_c$.
Although the arguments used by van den Berg and Brouwer look quite rigid, 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. | | Source: | arXiv, 0711.3563 | | Services: | Forum | Review | PDF | Favorites | |
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