Stretch-length exponent in spatial networks

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.

For $n$ distinct cities $z=(z_1,\ldots,z_n)$ in a square of area $n$, a Steiner network $N$ is a connected network of line segments joining them, with additional junctions allowed and crossings usable as junctions. Let $r_N(z_i,z_j)$ be shortest route length and \[S(N)=\max_{i\ne j}\frac{r_N(z_i,z_j)}{\|z_i-z_j\|_2},\qquad L(N)=\frac{\operatorname{len}(N)}n,\qquad \psi_n(z,s)=\inf_{S(N)\le s}L(N).\] Network length counts each segment once. For $Z_n$ consisting of $n$ independent uniform points in the area-$n$ square, define the established limits \[\Psi^{\rm ave}(s)=\lim_{n\to\infty}\mathbb E\psi_n(Z_n,s),\qquad\Psi^{\rm worst}(s)=\lim_{n\to\infty}\sup_z\psi_n(z,s)\quad(s>1).\] For positive functions $a,b$, $a\asymp b$ means their ratio stays between two fixed positive constants in the indicated limit.

Statement recorded in the supplied source.

Improve explicit upper and lower bounds for both functions. Is there an exponent $\alpha$ such that $\Psi(s)\asymp(s-1)^{-\alpha}$ as $s\downarrow1$ for each of $\Psi^{\rm ave}$ and $\Psi^{\rm worst}$? Determine whether a common exponent works for both settings. [S2]

Short English statement

Find sharper optimal network-length bounds under a stretch constraint and identify their divergence exponent as stretch approaches one.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9700018 (AMR-096-0018). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 David Aldous and Tamar Lando, The stretch-length tradeoff in geometric networks: average case and worst case study, Mathematical Proceedings of the Cambridge Philosophical Society 159 (2015), 125--145, Section 1, definitions and scaling question, pp. 1--3; Section 2, existence of limits. https://www.stat.berkeley.edu/~aldous/Papers/me142.pdf.