Optimal corrector estimates on percolation cluster

Abstract

We prove optimal quantitative estimates on the first-order correctors on supercritical percolation clusters: we show that they are bounded in dimension larger than 3 and have logarithmic growth in dimension 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 (Probab. Theory Related Fields 104 (1996) 427–466); large-scale regularity estimates developed by Armstrong and the author in (Comm. Pure Appl. Math. 71 (2018) 1717–1849); and a nonlinear concentration inequality of the Efron–Stein type which is used to transfer quantitative information from the environment to the correctors.