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The core of this note is the observation that links between circle packingsof graphs and potential theory developed in \cite{BeSc01} and \cite{HS} can beextended to higher dimensions. In particular, it is shown that every limit offinite graphs sphere packed in $\R^d$ with a uniformly-chosen root is$d$-parabolic. We then derive few geometric corollaries. E.g.\,every infinitegraph packed in $\R^{d}$ has either strictly positive isoperimetric Cheegerconstant or admits arbitrarily large finite sets $W$ with boundary size whichsatisfies $ |\partial W| \leq |W|^{\frac{d-1}{d}+o(1)}$. Some open problems andconjectures are gathered at the end.

On limits of Graphs Sphere Packed in Euclidean Space and Applications