Limit theorems for random spatial drainage networks

Penrose, Mathew D.; Wade, Andrew R.

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 Rd, d ≥ 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.

Citation

Penrose, M. D., & Wade, A. R. (2010). Limit theorems for random spatial drainage networks. Advances in Applied Probability, 42(3), 659-688. https://doi.org/10.1239/aap/1282924058

• Journal Article Type: Article
• Publication Date: Sep 1, 2010
• Deposit Date: Oct 4, 2012
• Publicly Available Date: Feb 13, 2013
• Journal: Advances in Applied Probability
• Print ISSN: 0001-8678
• Electronic ISSN: 1475-6064
• Publisher: Applied Probability Trust
• Peer Reviewed: Peer Reviewed
• Volume: 42
• Issue: 3
• Pages: 659-688
• DOI: https://doi.org/10.1239/aap/1282924058
• Keywords: Random spatial graph, Spanning tree, Weak convergence, Phase transition, Nearest-neighbour graph, Dickman distribution, Distributional fixed-point equation.
• Public URL: https://durham-repository.worktribe.com/output/1473333