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2014 | OriginalPaper | Buchkapitel

Cutting Edges at Random in Large Recursive Trees

verfasst von : Erich Baur, Jean Bertoin

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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.
Fußnoten
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.

Literatur
1.
L. Addario-Berry, N. Broutin, C. Holmgren, Cutting down trees with a Markov chainsaw. Ann. Appl. Probab. 24(6), 2297–2339 (2014)
2.
D. Aldous, The Continuum Random Tree III. Ann. Probab. 21, 248–289 (1993)
3.
A.-L. Barabási, R. Albert, Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)
4.
D. Barraez, S. Boucheron, W. de la Fernandez, Vega, On the fluctuations of the giant component. Comb. Probab. Comput. 9, 287–304 (2000)
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)
7.
J. Bertoin, Sizes of the largest clusters for supercritical percolation on random recursive trees. Random Struct. Algorithms 44–1, 1098–2418 (2014) MathSciNet
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
9.
J. Bertoin, The cut-tree of large recursive trees. To appear in Ann. Instit. Henri Poincaré Probab. Stat.
10.
J. Bertoin, G. Uribe Bravo, Supercritical percolation on large scale-free random trees. To appear in Ann. Appl. Probab.
11.
J. Bertoin, G. Miermont, The cut-tree of large Galton-Watson trees and the Brownian CRT. Ann. Appl. Probab. 23, 1469–1493 (2013)
12.
E. Bolthausen, A.-S. Sznitman, On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197–2, 247–276 (1998)
13.
B. Chauvin, T. Klein, J.-F. Marckert, A. Rouault, Martingales and profile of binary search trees. Electron. J. Probab. 10, 420–435 (2005)
14.
D. Dieuleveut,The vertex-cut-tree of Galton-Watson trees converging to a stable tree. To appear in Ann. Appl. Probab.
15.
M. Drmota, Random Trees (Springer, New York, Vienna, 2009)
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
17.
K.B. Erickson, Strong renewal theorems with infinite mean. Trans. Am. Math. Soc. 151, 263–291 (1970)
18.
C. Goldschmidt, J.B. Martin, Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Probab. 10, 718–745 (2005)
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)
20.
C. Holmgren, Random records and cuttings in binary search trees. Comb. Probab. Comput. 19–3, 391–424 (2010)
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)
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)
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
24.
S. Janson, Random cutting and records in deterministic and random trees. Random Struct. Algorithms 29–2, 139–179 (2006)
25.
O. Kallenberg, Foundations of Modern Probability, Probability and its Applications (Springer, New York, 2002) CrossRef
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)
27.
M. Kuba, A. Panholzer, Isolating nodes in recursive trees. Aequ. Math. 76, 258–280 (2008)
28.
M. Kuba, A. Panholzer, Multiple isolation of nodes in recursive trees. OJAC 9, 26 (2014)
29.
H. Mahmoud, Evolution of Random Search Trees (Wiley, New York, 1992) MATH
30.
H. Mahmoud, R.T. Smythe, A survey of recursive trees. Theor. Probab. Math. Stat. 51, 1–29 (1994)
31.
A. Meir, J.W. Moon, Cutting down random trees. J. Austral. Math. Soc. 11, 313–324 (1970)
32.
A. Meir, J.W. Moon, Cutting down recursive trees. Math. Biosci. 21, 173–181 (1974)
33.
A. Panholzer, Cutting down very simple trees. Quaest. Math. 29–2, 211–227 (2006)
34.
J. Pitman, Combinatorial Stochastic Processes. École d’été de Probabilités de St. Flour, Lecture Notes in Mathematics 1875 (Springer, Berlin, 2006)
35.
B. Pittel, On tree census and the giant component in sparse random graphs. Random Struct. Algorithms 1, 311–342 (1990)
36.
J. Schweinsberg, Dynamics of the evolving Bolthausen-Sznitman coalescent. Electron. J. Probab. 17, 1–50 (2012)
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)