Euclidean Matchings in Ultra-Dense Networks

In order to investigate the fundamental limits of network densification in and beyond 5G, we study the spatial spectral efficiency gain achieved when communication devices densely embedded in the d-dimensional Euclidean plane are optimally matched in near-neighbor pairs. We then proceed to assign these pairs their own data capacity given by Shannon's theorem. The length of the shortest matching on the points then corresponds to the maximum one-hop capacity in the network. Interference is then added as a further constraint, which is modeled using shapes as guard regions, such as a disk, diametral disk, or equilateral triangle, matched to points, in a similar manner 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.