Random recursive trees and the Bolthausen-Sznitman coalescent

Details Random recursive trees and the Bolthausen-Sznitman coalescent Random recursive trees and the Bolthausen-Sznitman coalescent

2005-02-13

arxiv.org/abs/math/0502263v2

Electron. J. Probab. Vol. 10 (2005) paper 21, pp. 718-745

Mathematics

Probability

28 pages, 2 figures. Revised version with minor alterations. To appear in Electron. J. Probab

Scientific paper

We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the final collision converges as n tends to infinity, and obtain a scaling law for the sizes of these blocks. We also consider the discrete-time Markov chain giving the number of blocks after each collision of the coalescent restricted to [n]; we show that the transition probabilities of the time-reversal of this Markov chain have limits as n tends to infinity. These results can be interpreted as describing a ``post-gelation'' phase of the Bolthausen-Sznitman coalescent, in which a giant cluster containing almost all of the mass has already formed and the remaining small blocks are being absorbed.

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