Random Recursive Trees and the Bolthausen-Sznitman Coalesent

@article{Goldschmidt2005RandomRT,
title={Random Recursive Trees and the Bolthausen-Sznitman Coalesent},
author={Christina Goldschmidt and James B. Martin},
journal={Electronic Journal of Probability},
year={2005},
volume={10},
pages={718-745}
}
• Published 13 February 2005
• Mathematics
• Electronic Journal of Probability
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…
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References

SHOWING 1-10 OF 43 REFERENCES
Coalescents with multiple collisions
k−2 � 1 − xb−k � � dx� . Call this process a � -coalescent. Discrete measure-valued processes derived from the � -coalescent model a system of masses undergoing coalescent collisions. Kingman's
Non-crossing trees revisited: cutting down and spanning subtrees
Two parameters for random non-crossing trees are considered: the number of random cuts to destroy a size-$n$ non-Crossing tree and the spanning subtree-size of randomly chosen nodes in a size-n non- crossing tree, and limiting distributions are obtained.
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
• Mathematics
• 1997
The two-parameter Poisson-Dirichlet distribution, denoted PD(α,θ), is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with
Destruction of Very Simple Trees
• Mathematics
Algorithmica
• 2006
We consider the total cost of cutting down a random rooted tree chosen from a family of so-called very simple trees (which include ordered trees, d-ary trees, and Cayley trees); these form a
The general coalescent with asynchronous mergers of ancestral lines
• S. Sagitov
• Mathematics
Journal of Applied Probability
• 1999
Take a sample of individuals in the fixed-size population model with exchangeable family sizes. Follow the ancestral lines for the sampled individuals backwards in time to observe the ancestral
Large deviation principles for some random combinatorial structures in population genetics and Brownian motion
• Mathematics
• 1998
Large deviation principles are established for some random combinatorial structures including the Ewens sampling formula and the Pitman sampling formula. A path-level large deviation principle is
A NECESSARY AND SUFFICIENT CONDITION FOR THE Λ-COALESCENT TO COME DOWN FROM INFINITY
Let Λ be a finite measure on the Borel subsets of [0, 1]. Let Π∞ be the standard Λ-coalescent, which is defined in [4] and also studied in [5]. Then Π∞ is a Markov process whose state space is the
Logarithmic Combinatorial Structures: A Probabilistic Approach
• Mathematics
• 2003
The elements of many classical combinatorial structures can be naturally decomposed into components. Permutations can be decomposed into cycles, polynomials over a finite field into irreducible
THE COALESCENT
• Mathematics
• 1980
The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes
• Mathematics
• 2000
Abstract. We use Bochner’s subordination to give a representation of the genealogical structure associated with general continuous-state branching processes. We then apply this representation to