The Subnetwork S(1) of a Scale-Invariant Random Spatial Network Is Not a Tree

D. Aldous asked for a proof that in a scale-invariant random spatial network (SIRSN) the subnetwork \(S(1)\), the union of the routes between the points of a Poisson process of rate one, cannot be a tree, even with Steiner points (Electron. J. Probab. 19 (2014), Open Problem 10; Open Problem 36 of the arXiv version). We prove this for every SIRSN and every weak SIRSN: the outcomes in which \(S(1)\) contains no simple closed curve form a set of outer probability zero, so that \(S(1)\) contains a circuit almost surely and, in particular, is not almost surely a tree. The same holds when “tree” means that the graph whose vertices are the Poisson points and the junctions of \(S(1)\) has no finite cycle. More generally, for any process of compatible routes with the invariance properties of a SIRSN, if that set has outer probability \(p>0\), then \(\mathbb{P}(D_1\ge t)\ge cp/t\) for all large \(t\), where \(D_1\) is the route length between two points at distance one and \(c>7.7\cdot10^{-11}\) is explicit; the SIRSN axiom \(\mathbb{E} D_1<\infty\) excludes this. The core of the proof is Aldous's argument for route lengths in invariant tree networks (Electron. Commun. Probab. 26 (2021)), whose percolation lemma is used as published. What the note adds is the transfer of this argument to the routes of a SIRSN, which need not consist of finitely many segments (a centroid bipartition and a route-length bound for finite unions of compatible routes), and the replacement of Palm probabilities by Campbell's formula and scale invariance. This is an unrefereed note.