Limit theorems for random spatial drainage networks

M.D. Penrose, A.R. Wade

Research output: Contribution to journalArticle

3 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)659-688
Number of pages29
JournalAdvances in Applied Probability
Volume42
Issue number3
DOIs
Publication statusPublished - 2010

Fingerprint

Spatial Networks
Random Networks
Limit Theorems
Drainage
Gravitation
Phase transitions
Convergence in Distribution
Liquids
Nearest Neighbor Graph
Network Evolution
Boundary Effect
Orthogonal Basis
Law of large numbers
Spanning tree
Regular hexahedron
Gravity
Phase Transition
Asymptotic Behavior
Non-negative
Liquid

Keywords

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

Cite this

Penrose, M.D. ; Wade, A.R. / Limit theorems for random spatial drainage networks. In: Advances in Applied Probability. 2010 ; Vol. 42, No. 3. pp. 659-688.
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Limit theorems for random spatial drainage networks. / Penrose, M.D.; Wade, A.R.

In: Advances in Applied Probability, Vol. 42, No. 3, 2010, p. 659-688.

Research output: Contribution to journalArticle

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AU - Penrose, M.D.

AU - Wade, A.R.

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