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New methods to bound the critical probability in fractal percolation
Author(s) -
Don Henk
Publication year - 2015
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.20566
Subject(s) - percolation (cognitive psychology) , fractal , percolation critical exponents , mathematics , upper and lower bounds , alphabet , statistical physics , percolation threshold , percolation theory , sequence (biology) , continuum percolation theory , random sequence , discrete mathematics , combinatorics , critical exponent , physics , topology (electrical circuits) , mathematical analysis , geometry , quantum mechanics , scaling , electrical resistivity and conductivity , linguistics , philosophy , genetics , neuroscience , biology , distribution (mathematics)
We study the critical probability p c ( M ) in two‐dimensional M ‐adic fractal percolation. To find lower bounds, we compare fractal percolation with site percolation. Fundamentally new is the construction of a computable increasing sequence that converges to p c ( M ). We prove thatp c ( 2 ) > 0.881 andp c ( 3 ) > 0.784 . For the upper bounds, we introduce an iterative random process on a finite alphabet A , which is easier to analyze than the original process. We show thatp c ( 2 ) ≤ 0.993 , p c ( 3 ) ≤ 0.940 andp c ( 4 ) ≤ 0.972 . © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 710–730, 2015