Rigidity of harmonic functions on the supercritical percolation cluster

Bou-Rabee, Ahmed; Cooperman, William; Dario, Paul

doi:10.1090/tran/9264

Rigidity of harmonic functions on the supercritical percolation cluster
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by Ahmed Bou-Rabee, William Cooperman and Paul Dario;

Trans. Amer. Math. Soc. 378 (2025), 3823-3896

DOI: https://doi.org/10.1090/tran/9264

Published electronically: March 19, 2025

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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.

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Rigidity of harmonic functions on the supercritical percolation cluster
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Abstract: