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

Route Lengths in Invariant Spatial Tree Networks