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Cutting Edges at Random in Large Recursive Trees

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Stochastic Analysis and Applications 2014

Part of the book series: Springer Proceedings in Mathematics & Statistics ((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.

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

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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.

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Correspondence to Jean Bertoin .

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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

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