Quantitative homogenization of the parabolic and elliptic Green’s functions on percolation clusters

Abstract

We study the heat kernel and the Green’s function on the infinite supercritical percolation cluster in dimension $\mathit{d}\ge 2$ and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly in (Electron. J. Probab. 14 (2009) 1–27). The proof relies on a structure of renormalization for the infinite percolation cluster introduced in (Comm. Pure Appl. Math. 71 (2018) 1717–1849), Gaussian bounds on the heat kernel established by Barlow in (Ann. Probab. 32 (2004) 3024–3084) and tools of the theory of quantitative stochastic homogenization. An important step in the proof is to establish a ${\mathit{C}}^{0,1}$-large-scale regularity theory for caloric functions on the infinite cluster and is of independent interest.