Quantitative homogenization and large-scale regularity of Poisson point clouds

TL;DR

This paper establishes quantitative homogenization and regularity results for harmonic functions on supercritical Poisson point clouds, demonstrating their large-scale Euclidean-like behavior with sharp bounds and error estimates.

Contribution

It provides the first quantitative homogenization results for harmonic functions on supercritical continuum percolation clusters, with optimal bounds and regularity estimates.

Findings

Harmonic functions on supercritical Poisson point clouds resemble Euclidean harmonic functions at large scales.

Sharp quantitative bounds on homogenization errors are established.

Large-scale regularity results are proven for the harmonic functions in the studied setting.

Abstract

We prove quantitative homogenization results for harmonic functions on supercritical continuum percolation clusters--that is, Poisson point clouds with edges connecting points which are closer than some fixed distance. We show that, on large scales, harmonic functions resemble harmonic functions in Euclidean space with sharp quantitative bounds on their difference. In particular, for every point cloud which is supercritical (meaning that the intensity of the Poisson process is larger than the critical parameter which guarantees the existence of an infinite connected component), we obtain optimal corrector bounds, homogenization error estimates and large-scale regularity results.