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This analysis demonstrates properties of the Green function in finitely generated groups, suggesting implications for random walks.
Given a probability measure μ on a finitely generated group Γ , the Green function G ( x , y | r ) encodes many properties of the random walk associated with μ . Endowing Γ with a word distance, we denote by H r ( n ) the sum of the Green function G ( e , x | r ) along the sphere of radius n . This quantity appears naturally when studying asymptotic properties of branching random walks driven by μ on Γ . Our main result exhibits a relatively hyperbolic group with convergent Poincaré series generated by H r ( n ) .
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Dussaule et al. (2025) studied this question.
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