Skip to main content Accessibility help
×
Home

Semi-Infinite Paths of the Two-Dimensional Radial Spanning Tree

  • François Baccelli (a1), David Coupier (a2) and Viet Chi Tran (a2)

Abstract

We study semi-infinite paths of the radial spanning tree (RST) of a Poisson point process in the plane. We first show that the expectation of the number of intersection points between semi-infinite paths and the sphere with radius r grows sublinearly with r. Then we prove that in each (deterministic) direction there exists, with probability 1, a unique semi-infinite path, framed by an infinite number of other semi-infinite paths of close asymptotic directions. The set of (random) directions in which there is more than one semi-infinite path is dense in [0, 2π). It corresponds to possible asymptotic directions of competition interfaces. We show that the RST can be decomposed into at most five infinite subtrees directly connected to the root. The interfaces separating these subtrees are studied and simulations are provided.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Semi-Infinite Paths of the Two-Dimensional Radial Spanning Tree
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Semi-Infinite Paths of the Two-Dimensional Radial Spanning Tree
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Semi-Infinite Paths of the Two-Dimensional Radial Spanning Tree
      Available formats
      ×

Copyright

Corresponding author

Postal address: Research group on Network Theory and Communications (TREC), INRIA-ENS, 75214 Paris, France.
∗∗ Postal address: Laboratoire Paul Painlevé, Université Lille 1, Cité Scientifique, 59655 Villeneuve d'Ascq Cedex, France.
∗∗∗ Email address: david.coupier@math.univ-lille1.fr

References

Hide All
[1] Athreya, S., Roy, R. and Sarkar, A. (2008). Random directed trees and forest-drainage networks with dependence. Electron. J. Prob. 13, 21602189.
[2] Baccelli, F. and Bordenave, C. (2007). The radial spanning tree of a Poisson point process. Ann. Appl. Prob. 17, 305359.
[3] Bonichon, N and Marckert, J.-F. (2011). Asymptotics of geometrical navigation on a random set of points in the plane. Adv. Appl. Prob. 43, 889942.
[4] Coupier, D. (2011). Multiple geodesics with the same direction. Electron. Commun. Prob. 16, 517527.
[5] Coupier, D. and Heinrich, P. (2011). Stochastic domination for the last passage percolation tree. Markov Process. Relat. Fields 17, 3748.
[6] Coupier, D. and Heinrich, P. (2012). Coexistence probability in the last passage percolation model is 6-8 log 2. Ann. Inst. H. Poincaré Prob. Statist. 48, 973988.
[7] Coupier, D. and Tran, V. C. (2013). The 2D-directed spanning forest is almost surely a tree. Random Structures Algorithms 42, 5972.
[8] Ferrari, P. A. and Pimentel, L. P. R. (2005). Competition interfaces and second class particles. Ann. Prob. 33, 12351254.
[9] Gangopadhyay, S., Roy, R. and Sarkar, A. (2004). Random oriented trees: a model of drainage networks. Ann. App. Prob. 14, 12421266.
[10] Howard, C. D. and Newman, C. M. (1997). Euclidean models of first-passage percolation. Prob. Theory Relat. Fields 108, 153170.
[11] Howard, C. D. and Newman, C. M. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Prob. 29, 577623.
[12] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
[13] Norris, J. and Turner, A. G. (2012). Hastings-Levitov aggregation in the small-particle limit. Commun. Math. Phys. 316, 809841.
[14] Pimentel, L. P. R. (2007). Multitype shape theorems for first passage percolation models. Adv. Appl. Prob. 39, 5376.
[15] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81, 73205.

Keywords

MSC classification

Semi-Infinite Paths of the Two-Dimensional Radial Spanning Tree

  • François Baccelli (a1), David Coupier (a2) and Viet Chi Tran (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed