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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 ..."
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Cited by 46 (3 self)
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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.
Smalltime behavior of beta coalescents
, 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 ..."
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Cited by 32 (12 self)
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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.
Asymptotic results concerning the total branch length of the Bolthausen–Sznitman coalescent
, 2007
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Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent, Ramanujan J. (to appear); Eprint
, 2005
"... Abstract. The twofold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and those of the second author for Euler’s constant γ and its alternating analog ln(4/π), and on the other hand the infinite products of the first author for e, and o ..."
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Abstract. The twofold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and those of the second author for Euler’s constant γ and its alternating analog ln(4/π), and on the other hand the infinite products of the first author for e, and of the second author for π and e γ. We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch’s transcendent of Hadjicostas’s double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions. The main tools are analytic continuations of Lerch’s function, including Hasse’s series. We also use Ramanujan’s polylogarithm formula for the sum of a certain series involving
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 ..."
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Cited by 14 (0 self)
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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.
A representation of exchangeable hierarchies by sampling from random real trees
, 2011
"... A hierarchy on a set S, also called a total partition of S, is a collection H of subsets of S such that S ∈ H, each singleton subset of S belongs to H, and if A,B ∈ H then A ∩ B equals either A or B or ∅. Every exchangeable random hierarchy of positive integers has the same distribution as a random ..."
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Cited by 7 (1 self)
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A hierarchy on a set S, also called a total partition of S, is a collection H of subsets of S such that S ∈ H, each singleton subset of S belongs to H, and if A,B ∈ H then A ∩ B equals either A or B or ∅. Every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy H associated as follows with a random real tree T equipped with root element 0 and a random probability distribution p on the Borel subsets of T: given (T, p), let t1, t2,... be independent and identically distributed according to p, and let H comprise all singleton subsets of N, and every subset of the form {j: tj ∈ Fx} as x ranges over T, where Fx is the fringe subtree of T rooted at x. There is also the alternative characterization: every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy H derived as follows from a random hierarchyH on [0, 1] and a family (Uj) of IID uniform [0,1] random variables independent ofH: let H comprise all sets of the form {j: Uj ∈ B} as B ranges over the members ofH.
Asymptotic results about the total branch length of the BolthausenSznitman coalescent
"... We study the total branch length Ln of the BolthausenSznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that Ln/E(Ln) converges to 1 in probability and that Ln, properly normalized, converges weakly to a stable random v ..."
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Cited by 6 (2 self)
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We study the total branch length Ln of the BolthausenSznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that Ln/E(Ln) converges to 1 in probability and that Ln, properly normalized, converges weakly to a stable random variable as n tends to infinity. The results are applied to derive a corresponding limiting law for the total number of mutations for the BolthausenSznitman coalescent with mutation rate r> 0. Moreover, the results show that, for the BolthausenSznitman coalescent, the total branch length Ln is closely related to Xn, the number of collision events that take place until there is just a single block. The proofs are mainly based on an analysis of random recursive equations using associated generating functions.
MULTIPLE ISOLATION OF NODES IN RECURSIVE TREES
"... ABSTRACT. We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate ℓ selected nodes in a size ..."
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ABSTRACT. We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate ℓ selected nodes in a sizen random recursive tree for three different selection rules, namely (i) isolating all of the nodes labelled 1, 2..., ℓ (thus nodes located close to the root of the tree), (ii) isolating all of the nodes labelled n + 1 − ℓ, n + 2 − ℓ,... n (thus nodes located at the fringe of the tree), and (iii) isolating ℓ nodes in the tree, which are selected at random before starting the edgeremoval procedure. Using a generating functions approach we determine for these selection rules the limiting distribution behaviour of the number of cuts to isolate all selected nodes, for ℓ fixed and n → ∞. 1.
On the length of an external branch in the betacoalescents
, 2012
"... In this paper, we consider Beta(2 − α,α) (with 1 < α < 2) and related Λcoalescents. If T (n) denotes the length of an external branch of the ncoalescent, we prove the convergence of n α−1 T (n) when n tends to ∞, and give the limit. To this aim, we give asymptotics for the number σ (n) of ..."
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Cited by 4 (3 self)
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In this paper, we consider Beta(2 − α,α) (with 1 < α < 2) and related Λcoalescents. If T (n) denotes the length of an external branch of the ncoalescent, we prove the convergence of n α−1 T (n) when n tends to ∞, and give the limit. To this aim, we give asymptotics for the number σ (n) of collisions which occur in the ncoalescent until the end of the chosen external branch, and for the block counting process associated with the ncoalescent.