We study the heat kernel and the Green’s function on the infinite supercritical percolation cluster in dimension 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 -large-scale regularity theory for caloric functions on the infinite cluster and is of independent interest.
We would like to thank Jean-Christophe Mourrat and Scott Armstrong for helpful discussions and comments. PD is supported by the Israel Science Foundation grants 861/15 and 1971/19 and by the European Research Council starting grant 678520 (LocalOrder). CG is supported by the PhD scholarship from École Polytechnique.
"Quantitative homogenization of the parabolic and elliptic Green’s functions on percolation clusters." Ann. Probab. 49 (2) 556 - 636, March 2021. https://doi.org/10.1214/20-AOP1456