Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-06T04:27:03.840Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)