  In order to study the fundamental limits of network densification, we look at
the spatial spectral efficiency gain achieved when densely deployed
communication devices embedded in the $d$-dimensional Euclidean space are
optimally `matched' in near-neighbour pairs. In light of recent success in
probabilisitc modelling, we study devices distributed uniformly at random in
the unit cube which enter into one-on-one contracts with each another. This is
known in statistical physics as an Euclidean `matching'. Communication channels
each have their own maximal data capacity given by Shannon's theorem. The
length of the shortest matching then corresponds to the maximum one-hop
capacity on those points. Interference is then added as a further constraint,
which is modelled using shapes as guard regions, such as a disk, diametral
disk, or equilateral triangle, matched to points, in a similar light to
computational geometry. The disk, for example, produces the Delaunay
triangulation, while the diametral disk produces a beta-skeleton. We also
discuss deriving the scaling limit of both models using the replica method from
the physics of disordered systems.
