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

Abstract

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 $ED(r,\theta )=O(r)$ 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.

References in corpus (1)

• Which Connected Spatial Networks on Random Points have Linear Route-Lengths?

Cited by in corpus (3)

• Connected Spatial Networks over Random Points and a Route-Length Statistic
• Scale-Invariant Random Spatial Networks
• Which Connected Spatial Networks on Random Points have Linear Route-Lengths?