  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.
