Limit theorems for random spatial drainage networks

TL;DR

This paper studies the asymptotic behavior of minimal-length drainage networks in a unit cube, revealing phase transitions and boundary effects, with results applicable to spatial network models and longest edge distributions.

Contribution

It provides new laws of large numbers and distributional convergence results for the total edge length and longest edge in random drainage networks, including phase transition phenomena.

Findings

Distributional limits show a phase transition between Gaussian and boundary-effect distributions.

Boundary effects are characterized via limits of the on-line nearest-neighbour graph.

Convergence in distribution for the longest edge length involves Dickman-type variables in 2D.

Abstract

Suppose that under the action of gravity, liquid drains through the unit $d$-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of $\R^d$, $d \geq 2$. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge-length of the network on uniform random points in $(0,1)^d$. The distributional results exhibit a weight-dependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when $d=2$, the limit is expressed in terms of Dickman-type variables.