Route Lengths in Invariant Spatial Tree Networks

TL;DR

This paper investigates the properties of invariant spatial tree networks connecting random points in the plane, focusing on the behavior of mean within-network distances as a function of Euclidean distance.

Contribution

It establishes a new result on the behavior of invariant tree networks in the continuum, extending previous lattice-based findings to the plane.

Findings

Mean within-network distance becomes infinite beyond a certain Euclidean distance

Provides a continuum analog of a known lattice result

Proves a weaker but related property for invariant tree networks

Abstract

Is there a constant $r_0$ such that, in any invariant tree network linking rate-$1$ Poisson points in the plane, the mean within-network distance between points at Euclidean distance $r$ is infinite for $r > r_0$? We prove a slightly weaker result. This is a continuum analog of a result of Benjamini et al (2001) on invariant spanning trees of the integer lattice.