# Limit theorems for random spatial drainage networks

### Reference:

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

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### Official URL:

http://dx.doi.org/10.1239/aap/1282924058

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- http://arxiv.org/abs/0901.3297

### 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 now with nonnegative component in one of the canonical orthogonal basis directions of R-d, 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.

### Details

Item Type | Articles | ||||||

Creators | Penrose, M. D.and Wade, A. R. | ||||||

DOI | 10.1239/aap/1282924058 | ||||||

Related URLs |
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Uncontrolled Keywords | distributional fixed-point equation,dickman distribution,random spatial graph,nearest-neighbour graph,spanning tree,phase transition,weak convergence | ||||||

Departments | Faculty of Science > Mathematical Sciences | ||||||

Research Centres | |||||||

Refereed | Yes | ||||||

Status | Published | ||||||

ID Code | 21323 |

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