Results 1  10
of
12
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
Abstract

Cited by 48 (3 self)
 Add to MetaCart
Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
SPINAL PARTITIONS AND INVARIANCE UNDER REROOTING OF CONTINUUM RANDOM TREES
, 2009
"... We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a twoparameter Poisson–Dirichlet family of continuous fragmentation trees ..."
Abstract

Cited by 25 (12 self)
 Add to MetaCart
(Show Context)
We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a twoparameter Poisson–Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform rerooting.
Asymptotics of the allele frequency spectrum associated with the BolthausenSznitman coalescent
, 2007
"... We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each individual, we trace back to the most recent mutation affecting it a ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each individual, we trace back to the most recent mutation affecting it and group together individuals whose most recent mutations are the same. The number of blocks of each of the different possible sizes in this partition is the allele frequency spectrum. The celebrated Ewens sampling formula gives precise probabilities for the allele frequency spectrum associated with Kingman’s coalescent. This (and the degenerate starshaped coalescent) are the only Λcoalescents for which explicit probabilities are known, although they are known to satisfy a recursion due to Möhle. Recently, Berestycki, Berestycki and Schweinsberg have proved asymptotic results for the allele frequency spectra of the Beta(2 − α,α) coalescents with α ∈ (1,2). In this paper, we prove full asymptotics for the case of the BolthausenSznitman coalescent.
Fragmentation of ordered partitions and intervals
 Electron. J. Probab
"... E l e c t r o n ..."
(Show Context)
Twoparameter PoissonDirichlet measures and reversible exchangeable fragmentationcoalescence processes
, 2007
"... ..."
Smalltime behavior of beta . . .
, 2008
"... For a finite measure Λ on [0, 1], the Λcoalescent is a coalescent process such that, whenever there are b clusters, each ktuple of clusters merges into one at rate ∫ 1 0 xk−2 (1 − x) b−k Λ(dx). It has recently been shown that if 1 < α < 2, the Λcoalescent in which Λ is the Beta(2−α, α) dist ..."
Abstract
 Add to MetaCart
For a finite measure Λ on [0, 1], the Λcoalescent is a coalescent process such that, whenever there are b clusters, each ktuple of clusters merges into one at rate ∫ 1 0 xk−2 (1 − x) b−k Λ(dx). It has recently been shown that if 1 < α < 2, the Λcoalescent in which Λ is the Beta(2−α, α) distribution can be used to describe the genealogy of a continuousstate branching process (CSBP) with an αstable branching mechanism. Here we use facts about CSBPs to establish new results about the smalltime asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other Λcoalescents for which Λ has the same asymptotic behavior near zero as the Beta(2 − α, α) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study smalltime properties of Λcoalescents.
Poisson Dirichlet(α, θ)Bridge Equations and CoagulationFragmentation Duality
, 2009
"... This paper derives distributional properties of a class of exchangeable bridges closely related to the PoissonDirichlet (α, θ) family of bridges. We then show that various stochastic equations derived for these bridges lead to constructions of a new large class of coagulation and fragmentation oper ..."
Abstract
 Add to MetaCart
(Show Context)
This paper derives distributional properties of a class of exchangeable bridges closely related to the PoissonDirichlet (α, θ) family of bridges. We then show that various stochastic equations derived for these bridges lead to constructions of a new large class of coagulation and fragmentation operators that satisfy a duality property, and are otherwise easily manipulated. This class, builds on, and includes the duality relations developed in Pitman (15), Bertoin and Goldschmidt (2), and Dong, Goldschmidt and Martin (4), which we can treat in a unified way. Our exposition also suggests an approach to obtain other dualities and related results.
Smalltime behavior . . .
, 2008
"... For a finite measure Λ on [0, 1], the Λcoalescent is a coalescent process such that, whenever there are b clusters, each ktuple of clusters merges into one at rate ∫ 1 0 xk−2 (1 − x) b−k Λ(dx). It has recently been shown that if 1 < α < 2, the Λcoalescent in which Λ is the Beta(2−α, α) dist ..."
Abstract
 Add to MetaCart
For a finite measure Λ on [0, 1], the Λcoalescent is a coalescent process such that, whenever there are b clusters, each ktuple of clusters merges into one at rate ∫ 1 0 xk−2 (1 − x) b−k Λ(dx). It has recently been shown that if 1 < α < 2, the Λcoalescent in which Λ is the Beta(2−α, α) distribution can be used to describe the genealogy of a continuousstate branching process (CSBP) with an αstable branching mechanism. Here we use facts about CSBPs to establish new results about the smalltime asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other Λcoalescents for which Λ has the same asymptotic behavior near zero as the Beta(2 − α, α) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study smalltime properties of Λcoalescents.