Skip to main content

Cutting Edges at Random in Large Recursive Trees

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 100)

Abstract

We comment on old and new results related to the destruction of a random recursive tree (RRT), in which its edges are cut one after the other in a uniform random order. In particular, we study the number of steps needed to isolate or disconnect certain distinguished vertices when the size of the tree tends to infinity. New probabilistic explanations are given in terms of the so-called cut-tree and the tree of component sizes, which both encode different aspects of the destruction process. Finally, we establish the connection to Bernoulli bond percolation on large RRT’s and present recent results on the cluster sizes in the supercritical regime.

Keywords

  • Random recursive tree
  • Destruction of graphs
  • Isolation of nodes
  • Disconnection
  • Supercritical percolation
  • Cluster sizes
  • Fluctuations

Acknowledgment of support    The research of the first author was supported by the Swiss National Science Foundation grant P2ZHP2_15640.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-11292-3_3
  • Chapter length: 26 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   129.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-11292-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   169.99
Price excludes VAT (USA)
Hardcover Book
USD   169.99
Price excludes VAT (USA)
Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    For the sake of simplicity, this notation does not record the order in which the edges are removed, although the latter is of course crucial in the definition of the cut-tree. In this part, we are concerned with uniform random edge removal, while in the last part of this section, we look at ordered destruction of a RRT, where edges are removed in the order of their endpoints most distant from the root.

References

  1. L. Addario-Berry, N. Broutin, C. Holmgren, Cutting down trees with a Markov chainsaw. Ann. Appl. Probab. 24(6), 2297–2339 (2014)

    Google Scholar 

  2. D. Aldous, The Continuum Random Tree III. Ann. Probab. 21, 248–289 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. A.-L. Barabási, R. Albert, Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)

    CrossRef  MathSciNet  Google Scholar 

  4. D. Barraez, S. Boucheron, W. de la Fernandez, Vega, On the fluctuations of the giant component. Comb. Probab. Comput. 9, 287–304 (2000)

    CrossRef  MATH  Google Scholar 

  5. E. Baur, Percolation on random recursive trees. Preprint (2014). arXiv:1407.2508

  6. J. Bertoin, Fires on trees. Ann. Inst. Henri Poincaré Probab. Stat. 48, 909–921 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. J. Bertoin, Sizes of the largest clusters for supercritical percolation on random recursive trees. Random Struct. Algorithms 44–1, 1098–2418 (2014)

    MathSciNet  Google Scholar 

  8. J. Bertoin, On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees. Electron. J. Probab. 19(24), 1–15 (2014)

    MathSciNet  Google Scholar 

  9. J. Bertoin, The cut-tree of large recursive trees. To appear in Ann. Instit. Henri Poincaré Probab. Stat.

    Google Scholar 

  10. J. Bertoin, G. Uribe Bravo, Supercritical percolation on large scale-free random trees. To appear in Ann. Appl. Probab.

    Google Scholar 

  11. J. Bertoin, G. Miermont, The cut-tree of large Galton-Watson trees and the Brownian CRT. Ann. Appl. Probab. 23, 1469–1493 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. E. Bolthausen, A.-S. Sznitman, On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197–2, 247–276 (1998)

    CrossRef  MathSciNet  Google Scholar 

  13. B. Chauvin, T. Klein, J.-F. Marckert, A. Rouault, Martingales and profile of binary search trees. Electron. J. Probab. 10, 420–435 (2005)

    CrossRef  MathSciNet  Google Scholar 

  14. D. Dieuleveut,The vertex-cut-tree of Galton-Watson trees converging to a stable tree. To appear in Ann. Appl. Probab.

    Google Scholar 

  15. M. Drmota, Random Trees (Springer, New York, Vienna, 2009)

    CrossRef  MATH  Google Scholar 

  16. M. Drmota, A. Iksanov, M. Möhle, U. Rösler, A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Struct. Algorithms 34–3, 319–336 (2009)

    CrossRef  Google Scholar 

  17. K.B. Erickson, Strong renewal theorems with infinite mean. Trans. Am. Math. Soc. 151, 263–291 (1970)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. C. Goldschmidt, J.B. Martin, Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Probab. 10, 718–745 (2005)

    CrossRef  MathSciNet  Google Scholar 

  19. B. Haas, G. Miermont, Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees. Ann. Probab. 40–6, 2589–2666 (2012)

    CrossRef  MathSciNet  Google Scholar 

  20. C. Holmgren, Random records and cuttings in binary search trees. Comb. Probab. Comput. 19–3, 391–424 (2010)

    CrossRef  MathSciNet  Google Scholar 

  21. C. Holmgren, A weakly 1-stable distribution for the number of random records and cuttings in split trees. Adv. Appl. Probab. 43–1, 151–177 (2011)

    CrossRef  MathSciNet  Google Scholar 

  22. A. Iksanov, M. Möhle, A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Comm. Probab. 12, 28–35 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. S. Janson, Random records and cuttings in complete binary trees, in Mathematics and Computer Science III, Algorithms, Trees, Combinatorics and Probabilities (Vienna 2004), ed. by M. Drmota, P. Flajolet, D. Gardy, B. Gittenberger (Basel, Birkhäuser, 2004), pp. 241–253

    Google Scholar 

  24. S. Janson, Random cutting and records in deterministic and random trees. Random Struct. Algorithms 29–2, 139–179 (2006)

    CrossRef  MathSciNet  Google Scholar 

  25. O. Kallenberg, Foundations of Modern Probability, Probability and its Applications (Springer, New York, 2002)

    CrossRef  Google Scholar 

  26. M. Kuba, A. Panholzer, On the degree distribution of the nodes in increasing trees. J. Combin. Theory, Series A 114–4, 597–618 (2007)

    Google Scholar 

  27. M. Kuba, A. Panholzer, Isolating nodes in recursive trees. Aequ. Math. 76, 258–280 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  28. M. Kuba, A. Panholzer, Multiple isolation of nodes in recursive trees. OJAC 9, 26 (2014)

    Google Scholar 

  29. H. Mahmoud, Evolution of Random Search Trees (Wiley, New York, 1992)

    MATH  Google Scholar 

  30. H. Mahmoud, R.T. Smythe, A survey of recursive trees. Theor. Probab. Math. Stat. 51, 1–29 (1994)

    MATH  MathSciNet  Google Scholar 

  31. A. Meir, J.W. Moon, Cutting down random trees. J. Austral. Math. Soc. 11, 313–324 (1970)

    CrossRef  MATH  MathSciNet  Google Scholar 

  32. A. Meir, J.W. Moon, Cutting down recursive trees. Math. Biosci. 21, 173–181 (1974)

    CrossRef  MATH  Google Scholar 

  33. A. Panholzer, Cutting down very simple trees. Quaest. Math. 29–2, 211–227 (2006)

    CrossRef  MathSciNet  Google Scholar 

  34. J. Pitman, Combinatorial Stochastic Processes. École d’été de Probabilités de St. Flour, Lecture Notes in Mathematics 1875 (Springer, Berlin, 2006)

    Google Scholar 

  35. B. Pittel, On tree census and the giant component in sparse random graphs. Random Struct. Algorithms 1, 311–342 (1990)

    CrossRef  MATH  MathSciNet  Google Scholar 

  36. J. Schweinsberg, Dynamics of the evolving Bolthausen-Sznitman coalescent. Electron. J. Probab. 17, 1–50 (2012)

    CrossRef  MathSciNet  Google Scholar 

  37. V.E. Stepanov, On the probability of connectedness of a random graph \(\cal {G}\) \(_m\) (t). Theory Probab. Appl. 15–1, 55–67 (1970)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jean Bertoin .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Baur, E., Bertoin, J. (2014). Cutting Edges at Random in Large Recursive Trees. In: Crisan, D., Hambly, B., Zariphopoulou, T. (eds) Stochastic Analysis and Applications 2014. Springer Proceedings in Mathematics & Statistics, vol 100. Springer, Cham. https://doi.org/10.1007/978-3-319-11292-3_3

Download citation