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In this paper, we provide for each $x\\in X$ and each $\\eta$ in the visual boundary $\\partial X$ of $X$ a description of the geodesic ray bundle $Geo(x,\\eta)$, namely, of the reunion of all combinatorial geodesic rays (corresponding to infinite minimal galleries in the chamber graph of $X$) starting from $x$ and pointing towards $\\eta$. When $X$ is locally finite and hyperbolic, we show that the symmetric difference between $Geo(x,\\eta)$ and $Geo(y,\\eta)$ is always finite, for $x,y\\in X$ and $\\eta\\in\\partial X$. 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