The Annals of Applied Probability

The radial spanning tree of a Poisson point process

Francois Baccelli and Charles Bordenave

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We analyze a class of spatial random spanning trees built on a realization of a homogeneous Poisson point process of the plane. This tree has a simple radial structure with the origin as its root.

We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of the edges or the mean degree of the vertices. Far away from the origin, these local properties are shown to be close to those of a variant of the directed spanning tree introduced by Bhatt and Roy.

We then use the theory of continuous state space Markov chains to analyze some nonlocal properties of the tree, such as the shape and structure of its semi-infinite paths or the shape of the set of its vertices less than k generations away from the origin.

This class of spanning trees has applications in many fields and, in particular, in communications.

Article information

Ann. Appl. Probab., Volume 17, Number 1 (2007), 305-359.

First available in Project Euclid: 13 February 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C05: Trees
Secondary: 90C27: Combinatorial optimization 60G55: Point processes

Spanning trees Poisson point process nearest neighbor graph directed spanning tree asymptotic shape


Baccelli, Francois; Bordenave, Charles. The radial spanning tree of a Poisson point process. Ann. Appl. Probab. 17 (2007), no. 1, 305--359. doi:10.1214/105051606000000826.

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  • Aldous, D. and Steele, M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopedia Math. Sci. 110 1–72. Springer, Berlin.
  • Alexander, K. (1995). Percolation and minimal spanning trees in infinite graphs. Ann. Probab. 23 87–104.
  • Athreya, K. and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 493–501.
  • Baccelli, F. and Brémaud, P. (2003). Elements of Queuing Theory, 2nd ed. Springer, Berlin.
  • Baccelli, F., Kofman, D. and Rougier, J. L. (1999). Self organizing hierarchical multicast trees and their optimization. In Proceedings of IEEE Infocom 3 1081–1089.
  • Bhatt, A. and Roy, R. (2004). On a random directed spanning tree. Adv. in Appl. Probab. 36 19–42.
  • Bordenave, C. (2006). Navigation on a Poisson point process. Rapport INRIA 5790. Available at
  • Daley, D. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications. Springer, New York.
  • Estrin, D., Govindan, R. and Heidemann, J (1999). Scalable coordination in sensor networks. Proceedings of ACM Mobicom 99 263–270.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications II, 2nd ed. Wiley, New York.
  • Ferrari, P., Landim, C. and Thorisson, H. (2004). Poisson trees, succession lines and coalescing random walks. Ann. Inst. H. Poincaré Probab. Statist. 40 141–152.
  • Ferrari, P., Fontes, L. R. and Wu, X.-Y. (2005). Two-dimensional Poisson trees converge to the Brownian web. Ann. Inst. H. Poincaré Probab. Statist. 41 851–858.
  • Fontes, L., Isopi, M., Newman, C. and Ravishankar, K. (2004). The Brownian web: Characterization and convergence. Ann. Probab. 32 2857–2883.
  • Gangopadhyay, S., Roy, R. and Sarkar, A. (2004). Random oriented trees: A model of drainage networks. Ann. Appl. Probab. 14 1242–1266.
  • Gut, A. (1980). Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices. Ann. Probab. 8 298–313.
  • Howard, C. and Newman, C. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Probab. 29 577–623.
  • Lee, S. (1997). The central limit theorem for Euclidian minimal spanning trees. Ann. Appl. Probab. 7 996–1020.
  • Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, Berlin.
  • Penrose, M. (2003). Random Geometric Graphs. Oxford Univ. Press.
  • Penrose, M. and Wade, A. (2004). Random minimal directed spanning trees and Dickman-type distributions. Adv. in Appl. Probab. 36 691–714.
  • Penrose, M. and Wade, A. (2006). On the total length of the random minimal directed spanning tree. Adv. in Appl. Probab. 38 336–372.
  • Penrose, M. and Yukich, J. (2001). Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 272–301.
  • Penrose, M. and Yukich, J. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277–303.
  • Rhee, W. (1993). A matching problem and subadditive Euclidean functionals. Ann. Appl. Probab. 3 794–801.
  • Steele, J. M. (1988). Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16 1767–1787.
  • Steele, J. M. (1997). Probability Theory and Combinatorial Optimization. SIAM, Philadelphia.
  • Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73–205.
  • Toth, B. and Werner, W. (1998). The true self repelling motion. Probab. Theory Related Fields 111 375–452.
  • Yukich, J. (1998). Probability Theory of Classical Euclidean Optimization Problems. Springer, Berlin.