The stretch-length tradeoff in spatial networks

Here is the abstract of Aldous-Lando (2015).
Consider a network linking the points of a rate-1 Poisson point process on the plane. Write \( \Psi^{ave}(s) \) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most \( s \) times the Euclidean distance. We give upper and lower bounds on the function \( \Psi^{ave}(s) \), and on the analogous "worst-case" function \( \Psi^{worst}(s) \) where the point configuration is arbitrary subject to average density one per unit area. Our bounds are numerically crude, but raise the question of whether there is an exponent \( \alpha \) such that each function has \( \Psi(s) \asymp (s-1)^{- \alpha} \mbox{ as } s \downarrow 1\).
One problem is to improve the explicit bounds: upper bounds require a construction, and lower bounds require some kind of "stochastic geometry" argument. Here is one improvement by Haoyu Wang. A more familiar kind of "theory" problem is to prove existence of the exponent \( \alpha \) in that dense network limit.

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