Shortest routes in random proximity networks

The class of proximity graphs provides a natural construction of connected spatial networks over arbitrary discrete vertex sets in the plane. Applied to a Poisson point process, one obtains a class of random proximity networks. Properties of such models have attracted comparatively little attention in mathematical probability. By analogy with first passage percolation, one can consider the shortest within-network route length $D_r$ between vertices at Euclidean distance $r$, and the transversal deviation $T_r$, that is the maximal deviation of the shortest route from the straight line. While there is a general SLLN for $D_r$, questions about the order of magnitude of $T_r$ and the variance of $D_r$ have apparently not been investigated before the 2019 simulation study by Kartun-Giles, Barthelemy and Dettmann. Rigorous study of such questions remains an open problem.