Random Recursive Trees and the Bolthausen-Sznitman Coalesent

By Christina Goldschmidt, James B. Martin. 2005

118 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ba088102efe23fffe128d1e286a830b8b8756fb4

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In Condensed Matter Physics

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Citations in the corpus

Combinatorial Stochastic Processes (2006, background): “See [72, 357, 54, 50, 176, 29] for various proofs of this fact, and more about this remarkable process.”
Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent (2005, methodology): “was first found by C. Goldschmidt and J. B. Martin [ 8 ] using probability theory.”
Small-time behavior of beta coalescents (2006, unknown): “ Beta(1,1) distribution is the uniform distribution on [0,1], and the associated coalescent process, called the Bolthausen–Sznitman coalescent, has already been studied extensively (see, for example, [2, 9, 12, 22, 30]). We focus here on the case in which 1<α<2. Some of our results are closely”
Recent progress in coalescent theory (2009, unknown): “y 161 recursive trees corresponds to the case where πis the trivial partition into singletons. With these deﬁnitions, we are able to state the following result, which is due to Goldschmidt and Martin [90], and which gives a striking construction of the Bolthausen-Sznitman coalescent in terms of rand”