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A SHORT PROOF OF THE HARRIS–KESTEN THEOREM

Published online by Cambridge University Press:  31 May 2006

BÉLA BOLLOBÁS  and
OLIVER RIORDAN
BÉLA BOLLOBÁS
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis TN 38152, USA and Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United [email protected], [email protected]
OLIVER RIORDAN
Affiliation:
Trinity College, Cambridge CB2 1TQ, United Kingdom and Royal Society Research Fellow, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United [email protected]
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Abstract

We give a short proof of the fundamental result that the critical probability for bond percolation in the planar square lattice ${\mathbb Z}^2$ 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 $p>1/2$.

Keywords

60K3582B43
Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 38 , Issue 3 , June 2006 , pp. 470 - 484
Copyright
© The London Mathematical Society 2006

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