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Linear lower bounds for δ c ( p ) for a class of 2D self‐destructive percolation models
Author(s) -
van den Berg J.,
de Lima B.N.B.
Publication year - 2009
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20265
Subject(s) - square lattice , combinatorics , mathematics , lattice (music) , hexagonal lattice , vertex (graph theory) , percolation (cognitive psychology) , percolation threshold , exponent , percolation theory , discrete mathematics , statistical physics , physics , condensed matter physics , topology (electrical circuits) , quantum mechanics , graph , linguistics , philosophy , neuroscience , antiferromagnetism , acoustics , ising model , electrical resistivity and conductivity , biology
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 δ to become occupied. Let δ c ( p ) be the minimal value of δ, 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 2 D lattices satisfying a special technical condition) that δ c ( p ) ≥ ${(p-p_{c})\over {p}} $ . In particular, δ 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 δ c ( p ) (with p near p c ) for a much larger class of 2 D lattices, including bond percolation on the square and triangular lattices, and site percolation on the star lattice (or matching lattice) of the square lattice. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009