Asymptotic properties of Euclidean shortest-path trees in random geometric graphs

We consider asymptotic properties of two functionals on Euclidean shortest-path trees appearing in random geometric graphs in R2 which can be used, for example, as models for fixed-access telecommunication networks. First, we determine the asymptotic bivariate distribution of the two backbone lengths inside a certain class of typical Cox-Voronoi cells as the size of this cell grows unboundedly. The corresponding Voronoi tessellation is generated by a stationary Cox process which is concentrated on the edges of the random geometric graph and whose intensity tends to 0. The limiting random vector can be represented as a simple geometric functional of a decomposition of a typical Poisson-Voronoi cell induced by an independent random sector. Using similar methods, we consider the asymptotic bivariate distribution of the total lengths of the two subtrees inside the Cox-Voronoi cell.