Route lengths in invariant spatial tree networks

Abstract

Is there a constant r0 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>r0? 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.