Rigidity of harmonic functions on the supercritical percolation cluster

Abstract

We use ideas from quantitative homogenization to show that nonconstant harmonic functions on the percolation cluster cannot satisfy certain structural constraints, for example, a Lipschitz bound. These unique-continuation-type results are false on the full lattice and hence the disorder is utilized in an essential way.

arXiv comment73 pages, 25 figures; v2 minor revisions