LIMIT THEOREMS FOR RANDOM SPATIAL DRAINAGE NETWORKS

作者：Penrose Mathew D^*; Wade Andrew R

来源：Advances in Applied Probability, 2010, 42(3): 659-688.

• Random spatial graph
• spanning tree
• weak convergence
• phase transition
• nearest-neighbour graph
• Dickman distribution
• distributional fixed-point equation

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.

• 出版日期2010-9