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Some remarks concerning harmonic functions on homogeneous graphs

Abstract : We obtain a new result concerning harmonic functions on infinite Cayley graphs $X$: either every nonconstant harmonic function has infinite radial variation in a certain uniform sense, or there is a nontrivial boundary with hyperbolic properties at infinity of $X$. In the latter case, relying on a theorem of Woess, it follows that the Dirichlet problem is solvable with respect to this boundary. Certain relations to group cohomology are also discussed.
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Anders Karlsson. Some remarks concerning harmonic functions on homogeneous graphs. Discrete Random Walks, DRW'03, 2003, Paris, France. pp.137-144, ⟨10.46298/dmtcs.3348⟩. ⟨hal-01183944⟩

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