Limit theorems for random spatial drainage networks

Details Limit theorems for random spatial drainage networks Limit theorems for random spatial drainage networks

2009-01-21

arxiv.org/abs/0901.3297v1

Advances in Applied Probability, Vol. 42 (2010), no. 3, p. 659-688

Mathematics

Probability

33 pages, 1 colour figure

Scientific paper

10.1239/aap/1282924058

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 $\R^d$, $d \geq 2$. 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.

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