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Cutting edges at random in large recursive trees
, 2014
"... 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 tr ..."
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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 socalled cuttree 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. Key words: Random recursive tree, destruction of graphs, isolation of nodes, disconnection, supercritical percolation, cluster sizes, fluctuations. 1
Percolation on random recursive trees
, 2014
"... We study Bernoulli bond percolation on a random recursive tree of size n with percolation parameter p(n) converging to 1 as n tends to infinity. The sizes of the percolation clusters are naturally stored in a tree. We prove convergence in distribution of this tree to the genealogical tree of a cont ..."
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We study Bernoulli bond percolation on a random recursive tree of size n with percolation parameter p(n) converging to 1 as n tends to infinity. The sizes of the percolation clusters are naturally stored in a tree. We prove convergence in distribution of this tree to the genealogical tree of a continuousstate branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin [5] which deals with cluster sizes in the supercritical regime. In a second part, we show that the same limit tree appears in the study of the tree components which emerge from a continuoustime destruction of a random recursive tree. We comment on the connection to our first result on Bernoulli bond percolation.
Reversing the cut tree of the Brownian continuum random tree
, 2014
"... Consider the logging process of the Brownian continuum random tree (CRT) T using a Poisson point process of cuts on its skeleton [Aldous and Pitman, Ann. Probab., vol. 26, pp. 1703–1726, 1998]. Then, the cut tree introduced by Bertoin and Miermont describes the genealogy of the fragmentation of T i ..."
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Consider the logging process of the Brownian continuum random tree (CRT) T using a Poisson point process of cuts on its skeleton [Aldous and Pitman, Ann. Probab., vol. 26, pp. 1703–1726, 1998]. Then, the cut tree introduced by Bertoin and Miermont describes the genealogy of the fragmentation of T into connected components [Ann. Appl. Probab., vol. 23, pp. 1469–1493, 2013]. This cut tree cut(T) is distributed as another Brownian CRT, and is a function of the original tree T and of the randomness in the logging process. We are interested in reversing the transformation of T into cut(T): we define a shuffling operation, which given a Brownian CRT H, yields another one shuff(H) distributed in such a way that (T, cut(T)) and (shuff(H),H) have the same distribution. 1