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Disorder, entropy and harmonic functions

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Since the work of Yau in 1975, where a Liouville property for positive harmonic functions on complete manifolds with non-negative Ricci curvature is proved, the structure of various spaces of harmonic functions have been at the heart of geometric analysis. We extend this question to the random context, with an emphasize on the infinite cluster of percolation. We will prove that for almost every supercritical-cluster of percolation, there are no non-constant sublinear harmonic functions. The main ingredient of the proof is a quantitative, annealed version of the Kaimanovich-Vershik entropy argument. We will also mention several open problems and conjectures on the behavior of harmonic functions on stationary random graphs.

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