A short proof of the Harris-Kesten theorem
We give a short proof of the fundamental result that the critical probability for bond percolation in the planar square lattice ℤ 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. © 2006 London Mathematical Society.
Bulletin of the London Mathematical Society
Bollobás, B., & Riordan, O. (2006). A short proof of the Harris-Kesten theorem. Bulletin of the London Mathematical Society, 38 (3), 470-484. https://doi.org/10.1112/S002460930601842X