Percolation in High Dimensions
In 1990 Kesten  proved that the critical probability pc (Zn, site) for site percolation in Zn is at most (1 + O((log log n )2/log n))/2n. Together with the immediate lower bound of 1/(2n - 1), this result shows that pc (Zn, site) = (1 + o (1))/2n. Since the critical probability pc (Zn, bond) for bond percolation in Zn is no greater than pc (Zn, site), and pc (Zn, bond) ≥ 1/(2n - 1) as well, we have that pc(Zn, bond) = (1 + o(1))/2n also holds (see also Gordon ). In a remarkable paper , in which Hara and Slade prove that the so-called triangle condition holds in certain percolation processes in Zn, it is shown that, in fact, pc(Zn, bond) = (1 + O(1/n))/2n. In the same paper Hara and Slade also state in passing that their methods give pc (Zn, site) = (1 + O (1/n))/2n. The main aim of this note is to give a self-contained simple proof of the inequality pc(Zn, site) ≤ (1 + no(1)-1/3 )/2n. Our methods differ greatly from those of Kesten and of Hara and Slade; in particular, our proofs are entirely combinatorial. We then estimate pc (Zn, bond) using a variant of our method, and give a simple proof of a result that is only slightly weaker than that of Hara and Slade, namely that pc (Zn, bond) ≤ (1 + O ((log n)2/n))/2n. © 1994 Academic Press. All rights reserved.
European Journal of Combinatorics
Bollobás, B., & Kohayakawa, Y. (1994). Percolation in High Dimensions. European Journal of Combinatorics, 15 (2), 113-125. https://doi.org/10.1006/eujc.1994.1014