Electronic Journal of Probability

Random Recursive Trees and the Bolthausen-Sznitman Coalesent

Christina Goldschmidt and James Martin

Full-text: Open access

Abstract

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.

Article information

Source
Electron. J. Probab. Volume 10 (2005), paper no. 21, 718-745.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464816822

Digital Object Identifier
doi:10.1214/EJP.v10-265

Mathematical Reviews number (MathSciNet)
MR2164028

Zentralblatt MATH identifier
1109.60060

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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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