Is there a constant 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 ? 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.
I thank Yuval Peres and Russ Lyons for the references to [10, 15], and Geoffrey Grimmett for comments on the contour method and for catching an error in an early draft. I also thank an anonymous referee for prompting some more detailed proofs.
"Route lengths in invariant spatial tree networks." Electron. Commun. Probab. 26 1 - 12, 2021. https://doi.org/10.1214/21-ECP401