  We prove that quadratic pair interactions for functions defined on planar
Poisson clouds and taking into account pairs of sites of distance up to a
certain (large-enough) threshold can be almost surely approximated by the
multiple of the Dirichlet energy by a deterministic constant. This is achieved
by scaling the Poisson cloud and the corresponding energies and computing a
compact discrete-to-continuum limit. In order to avoid the effect of
exceptional regions of the Poisson cloud, with an accumulation of sites or with
"disconnected sites", a suitable "coarse-grained" notion of convergence of
functions defined on scaled Poisson clouds must be given.
