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