Electronic Journal of Probability

Random Recursive Trees and the Bolthausen-Sznitman Coalesent

Christina Goldschmidt and James Martin

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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 $n\to\infty$, 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 $n\to\infty$. 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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Electron. J. Probab., Volume 10 (2005), paper no. 21, 718-745.

Accepted: 14 July 2005
First available in Project Euclid: 1 June 2016

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Goldschmidt, Christina; Martin, James. Random Recursive Trees and the Bolthausen-Sznitman Coalesent. Electron. J. Probab. 10 (2005), paper no. 21, 718--745. doi:10.1214/EJP.v10-265. https://projecteuclid.org/euclid.ejp/1464816822

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