  We prove optimal quantitative estimates on the first-order correctors on
supercritical percolation clusters: we show that they are bounded in $d\geq 3$
and have logarithmic growth in $d = 2$, in the sense of stretched exponential
moments. The main ingredients are a renormalization scheme of the supercritical
percolation cluster, following the works of Pisztora and Barlow; large-scale
regularity estimates developed in the previous paper; and a nonlinear
concentration inequality of Efron-Stein type which is used to transfer
quantitative information from the environment to the correctors.
