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We study the heat kernel and the Green's function on the infinitesupercritical percolation cluster in dimension $d \geq 2$ and prove aquantitative homogenization theorem for these functions with an almost optimalrate of convergence. These results are a quantitative version of the localcentral limit theorem proved by Barlow and Hambly. The proof relies on astructure of renormalization for the infinite percolation cluster introduced byArmstrong and the first author, Gaussian bounds on the heat kernel establishedby Barlow and tools of the theory of quantitative stochastic homogenization. Animportant step in the proof is to establish a $C^{0,1}$-large-scale regularitytheory for caloric functions on the infinite cluster and is of independentinterest.

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