The Shape Theorem for Route-lengths in Connected Spatial Networks on Random Points

For a connected network on Poisson points in the plane, consider the route-length $D(r,\theta) $ between a point near the origin and a point near polar coordinates $(r,\theta)$, and suppose $E D(r,\theta) = O(r)$ as $r \to \infty$. By analogy with the shape theorem for first-passage percolation, for a translation-invariant and ergodic network one expects $r^{-1} D(r, \theta)$ to converge as $r \to \infty$ to a constant $\rho(\theta)$. It turns out there are some subtleties in making a precise formulation and a proof. We give one formulation and proof via a variant of the subadditive ergodic theorem wherein random variables are sometimes infinite.