Advances in Applied Probability

Random minimal directed spanning trees and Dickman-type distributions

Mathew D. Penrose and Andrew R. Wade

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In Bhatt and Roy's minimal directed spanning tree construction for n random points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for large n) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

Article information

Adv. in Appl. Probab. Volume 36, Number 3 (2004), 691-714.

First available in Project Euclid: 31 August 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G70: Extreme value theory; extremal processes
Secondary: 05C80: Random graphs [See also 60B20] 60F05: Central limit and other weak theorems

Spanning tree extreme value weak convergence Dickman distribution Poisson-Dirichlet distribution


Penrose, Mathew D.; Wade, Andrew R. Random minimal directed spanning trees and Dickman-type distributions. Adv. in Appl. Probab. 36 (2004), no. 3, 691--714. doi:10.1239/aap/1093962229.

Export citation


  • Arnold, B. C. and Villaseñor, J. A. (1998). The asymptotic distribution of sums of records. Extremes 1, 351--363.
  • Arratia, R. (1998). On the central role of scale invariant Poisson processes on $(0,\infty)$. In Microsurveys in Discrete Probability (Princeton, NJ, 1997; DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 41), American Mathematical Society, Providence, RI, pp. 21--41.
  • Barndorff-Nielsen, O. and Sobel, M. (1966). On the distribution of the number of admissible points in a vector random sample. Theory Prob. Appl. 11, 249--269.
  • Bhatt, A. G. and Roy, R. (2004). On a random directed spanning tree. Adv. Appl. Prob. 36, 19--42.
  • Devroye, L. and Neininger, R. (2002). Density approximation and exact simulation of random variables that are solutions of fixed-point equations. Adv. Appl. Prob. 34, 441--468.
  • Dickman, K. (1930). On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astronom. Fys. 22A, No. 10, 1--14.
  • Donnelly, P. and Grimmett, G. (1993). On the asymptotic distribution of large prime factors. J. London Math. Soc. 47, 395--404.
  • Durrett, R. (1991). Probability: Theory and Examples. Wadsworth and Brooks, Pacific Grove, CA.
  • Goldie, C. M. and Grübel, R. (1996). Perpetuities with thin tails. Adv. Appl. Prob. 28, 463--480.
  • Griffiths, R. C. (1988). On the distribution of points in a Poisson process. J. Appl. Prob. 25, 336--345.
  • Hensley, D. (1986). The convolution powers of the Dickman function. J. London Math. Soc. 33, 395--406.
  • Holst, L. (2001). The Poisson--Dirichlet distribution and its relatives revisited. Preprint, Royal Institute of Technology, Stockholm. Available at
  • Hwang, H.-K. and Tsai, T.-H. (2002). Quickselect and Dickman function. Combin. Prob. Comput. 11, 353--371.
  • Kingman, J. F. C. (1993). Poisson Processes (Oxford Stud. Prob. 3). Oxford University Press.
  • Kolmogorov, A. N. and Fom\=\i n, S. V. (1975). Introductory Real Analysis. Dover, New York.
  • Kruskal, J. B. (1956). On the shortest spanning subtree of a graph and the travelling salesman problem. Proc. Amer. Math. Soc. 7, 48--50.
  • Mahmoud, H. M. and Smythe, R. T. (1998). Probabilistic analysis of MULTIPLE QUICK SELECT. Algorithmica 22, 569--584.
  • Penrose, M. (2003). Random Geometric Graphs (Oxford Stud. Prob. 5). Oxford University Press.
  • Penrose, M. D. and Wade, A. R. (2004). On the total length of the random minimal directed spanning tree. In preparation.
  • Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Prob. 11, 1005--1041.
  • Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277--303.
  • Rodriguez-Iturbe, I. and Rinaldo, A. (1997). Fractal River Basins: Chance and Self-Organization. Cambridge University Press.
  • Steele, J. M. (1997). Probability Theory and Combinatorial Optimization. Society for Industrial and Applied Mathematics, Philadelphia, PA.
  • Tenenbaum, G. (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press.
  • Watterson, G. A. (1976). The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Prob. 13, 639--651.
  • Yukich, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems (Lecture Notes Math. 1675). Springer, Berlin.