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Limit theorems for random spatial drainage networks.

Penrose, Mathew D. and Wade, Andrew R. (2010) 'Limit theorems for random spatial drainage networks.', Advances in Applied Probability, 42 (3). pp. 659-688.

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.

Item Type:Article
Keywords:Random spatial graph, Spanning tree, Weak convergence, Phase transition, Nearest-neighbour graph, Dickman distribution, Distributional fixed-point equation.
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1239/aap/1282924058
Date accepted:No date available
Date deposited:13 February 2013
Date of first online publication:September 2010
Date first made open access:No date available

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