Limit theorems for random spatial drainage networks
Penrose, M.D. and Wade, A.R. (2010) Limit theorems for random spatial drainage networks. Advances in Applied Probability, 42 (3). pp. 659688. ISSN 00018678 (http://projecteuclid.org/euclid.aap/1282924058)
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Suppose that, under the action of gravity, liquid drains through the unit dcube via a minimallength 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 socalled minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the largesample asymptotic behaviour of the total powerweighted edge length of the network on uniform random points in (0, 1)d. The distributional results exhibit a weightdependent phase transition between Gaussian and boundaryeffectderived distributions. These boundary contributions are characterized in terms of limits of the socalled online nearestneighbour 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 Dickmantype variables.


Item type: Article ID code: 27356 Dates: DateEvent2010PublishedKeywords: random spatial graph, spanning tree, weak convergence, phase transition, nearestneighbour graph, Dickman distribution, distributional fixedpoint equation, Mathematics, Applied Mathematics, Statistics and Probability Subjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Mrs Carolynne Westwood Date deposited: 03 Sep 2010 14:12 Last modified: 02 Apr 2021 03:02 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/27356