Mathew D. Penrose
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 | Mar 29, 2024 |
Journal | Advances in Applied Probability |
Print ISSN | 0001-8678 |
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. |
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