Abstract
We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on $\mathbb{Z}^{d}$. We prove that the vector space of harmonic functions growing at most linearly is $(d+1)$-dimensional almost surely. Further, there are no nonconstant sublinear harmonic functions (thus implying the uniqueness of the corrector). A main ingredient of the proof is a quantitative, annealed version of the Avez entropy argument. This also provides bounds on the derivative of the heat kernel, simplifying and generalizing existing results. The argument applies to many different environments; even reversibility is not necessary.
Citation
Itai Benjamini. Hugo Duminil-Copin. Gady Kozma. Ariel Yadin. "Disorder, entropy and harmonic functions." Ann. Probab. 43 (5) 2332 - 2373, September 2015. https://doi.org/10.1214/14-AOP934
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