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A short proof of the Harris-Kesten theorem. (English) Zbl 1104.60060
A fundamental statement in percolation theory asserts that the critical probability for bond percolation in the planar square lattice \(\mathbb{Z}^2\) is equal to 1/2. The lower bond was proved by Harris, who showed in 1960 that percolation does not occur at \(p=1/2\). The other bound was proved by Kesten, who showed in 1980 that percolation does occur for any \(p>1/2\). The authors give here short and elegant proofs of both results. A general result of probabilistic combinatorics, due to E. Friedgut and G. Kalai [Proc. Am. Math. Soc. 124, 2993–3002 (1996; Zbl 0864.05078)], is recalled and three ways of deducing Kesten’s theorem are given. The same method gives similar results in other contexts than bond percolation.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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