Optimal corrector estimates on percolation clusters

Abstract

We prove optimal quantitative estimates on the first-order correctors on supercritical percolation clusters: we show that they are bounded in $d\ge 3$ 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.

51 pages; revised version, accepted for publication in Annals of Applied Probability