  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 $d\ge2$.
