Expected length of a SIRSN spanning subnetwork

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A scale-invariant random spatial network (SIRSN) is a jointly specified family of random routes $R(z,z')$ between points of $\mathbb R^2$. Routes are non-self-intersecting curves of finite length made of straight segments, with finitely or countably many segments whose only possible limiting endpoints are $z,z'$. The reverse route is $R(z',z)$. Any two routes meeting twice coincide between those meeting points. For every finite set of endpoints, the spanning subnetwork is the union of all its pairwise routes. These finite-dimensional laws are consistent under restriction, measurable in the endpoints, and invariant under translations, rotations and simultaneous multiplication of all coordinates by any $c>0$. Let $\Xi(\lambda)$ be independent planar Poisson points of intensity $\lambda$, coupled increasingly in $\lambda$ by Poisson rain, and let $S(\lambda)$ be the union of the routes between their pairs. These subnetworks have almost surely finite length in every bounded set. For a stationary edge set $F$, its edge intensity $\operatorname{int}(F)$ is defined by $\mathbb E\operatorname{len}(F\cap A)=\operatorname{int}(F)\operatorname{area}(A)$ for bounded Borel sets $A$; length counts each road segment only once. Write $B(z,r)$ for a Euclidean disc, $e=(1,0)$, and \[ D_1=\operatorname{len}R(0,e),\quad \Delta=\mathbb E D_1,\quad \ell=\operatorname{int}S(1), \] \[ E(\lambda,r)=\bigcup_{\xi,\xi'\in\Xi(\lambda)} \left(R(\xi,\xi')\setminus(B(\xi,r)\cup B(\xi',r))\right),\quad E(\infty,r)=\bigcup_{\lambda>0}E(\lambda,r),\quad p(r)=\lim_{\lambda\to\infty}\operatorname{int}E(\lambda,r). \] The SIRSN assumptions include $1<\Delta<\infty$ and $p(1)<\infty$; they imply $\ell<\infty$. A weak SIRSN replaces the requirement $p(1)<\infty$ by $\ell<\infty$. Routes are given primitives and need not be shortest paths for Euclidean length. For each integer $k\ge1$, take $Z_1,\ldots,Z_k$ independent uniform points in a square of area $k$, independent of the network, and put \[\operatorname{span}(Z_1,\ldots,Z_k)=\bigcup_{1\le i<j\le k}R(Z_i,Z_j).\] Its length counts each segment once and includes portions lying outside the square.

Statement recorded in the supplied source.

Prove, for every SIRSN, that $\mathbb E\operatorname{len}[\operatorname{span}(Z_1,\ldots,Z_k)]\sim\ell k$ as $k\to\infty$, where $\sim$ means that the ratio tends to one. [S2]

Short English statement

Does the whole network joining uniformly sampled points in an expanding square have expected length asymptotic to its intensity times the square area?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9700035 (AMR-096-0035). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 David J. Aldous, Scale-Invariant Random Spatial Networks, arXiv:1204.0817v1 (2012), Section 8.4.3, Open Problem 35, p. 50. https://arxiv.org/pdf/1204.0817.