  We study the heat kernel and the Green's function on the infinite
supercritical percolation cluster in dimension $d \geq 2$ 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. The proof relies on a
structure of renormalization for the infinite percolation cluster introduced by
Armstrong and the first author, Gaussian bounds on the heat kernel established
by Barlow and tools of the theory of quantitative stochastic homogenization. An
important step in the proof is to establish a $C^{0,1}$-large-scale regularity
theory for caloric functions on the infinite cluster and is of independent
interest.
