Upper bounds on the percolation correlation length

Abstract

We study the size of the near-critical window for Bernoulli percolation on ${\mathbb{Z}}^d$. More precisely, we use a quantitative Grimmett-Marstrand theorem to prove that the correlation length, both below and above criticality, is bounded from above by $\exp (C/|p-p_c{|}^2)$. Improving on this bound would be a further step towards the conjecture that there is no infinite cluster at criticality on ${\mathbb{Z}}^d$ for every $d\ge 2$.

21 pages, 2 figures