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A new lower bound for the critical probability of site percolation on the square lattice
Author(s) -
van den Berg J.,
Ermakov A.
Publication year - 1996
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/(sici)1098-2418(199605)8:3<199::aid-rsa4>3.0.co;2-t
Subject(s) - square lattice , upper and lower bounds , combinatorics , mathematics , square (algebra) , percolation (cognitive psychology) , percolation critical exponents , percolation threshold , lattice (music) , statistical physics , directed percolation , discrete mathematics , critical exponent , physics , mathematical analysis , quantum mechanics , geometry , ising model , scaling , neuroscience , acoustics , biology , electrical resistivity and conductivity
The critical probability for site percolation on the square lattice is not known exactly. Several authors have given rigorous upper and lower bounds. Some recent lower bounds are (each displayed here with the first three digits) 0.503 (Tóth [13]), 0.522 (Zuev [15]), and the best lower bound so far, 0.541 (Menshikov and Pelikh [12]). By a modification of the method of Menshikov and Pelikh we get a significant improvement, namely, 0.556. Apart from a few classical results on percolation and coupling, which are explicitly stated in the Introduction, this paper is self‐contained. © 1996 John Wiley & Sons, Inc.