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A Short Proof of the Harris–Kesten Theorem
Author(s) -
Bollobás Béla,
Riordan Oliver
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s002460930601842x
Subject(s) - mathematics , percolation (cognitive psychology) , combinatorics , square lattice , upper and lower bounds , planar , lattice (music) , discrete mathematics , mathematical physics , statistical physics , mathematical analysis , physics , ising model , computer graphics (images) , neuroscience , computer science , acoustics , biology
We give a short proof of the fundamental result that the critical probability for bond percolation in the planar square lattice Z 2 is equal to 1/2. The lower bound was proved by Harris, who showed in 1960 that percolation does not occur at p = 1/2. The other, more difficult, bound was proved by Kesten, who showed in 1980 that percolation does occur for any p > 1/2. 2000 Mathematics Subject Classification 60K35, 82B43.