## 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.