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Following Kaimanovich's elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure ${\\mathfrak E}$ exists; it is hyperbolic, and the hyperbolic boundary $\\partial_HX$ with the Gromov metric is H\\\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0< \\lambda <1$ on $(X, {\\mathfrak E})$. We show that the Martin boundary ${\\mathcal M}$ can be identified with $\\partial_H X$ and $K$. 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