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 $d\ge 2$ 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. 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.

73 pages, 6 figures

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• Lower Gaussian heat kernel bounds for the Random Conductance Model in a degenerate ergodic environment
• Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian
• First passage percolation with long-range correlations and applications to random Schrödinger operators
• Absence of weak disorder for directed polymers on supercritical percolation clusters
• Coupling between Brownian motion and random walks on the infinite percolation cluster