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111
Coalescents With Multiple Collisions
 Ann. Probab
, 1999
"... For each finite measure on [0 ..."
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A classification of coalescent processes for haploid exchangeable population models
 Ann. Probab
, 2001
"... We consider a class of haploid population models with nonoverlapping generations and fixed population size N assuming that the family sizes within a generation are exchangeable random variables. A weak convergence criterion is established for a properly scaled ancestral process as N! 1. It results ..."
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Cited by 63 (4 self)
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We consider a class of haploid population models with nonoverlapping generations and fixed population size N assuming that the family sizes within a generation are exchangeable random variables. A weak convergence criterion is established for a properly scaled ancestral process as N! 1. It results in a full classification of the coalescent generators in the case of exchangeable reproduction. In general the coalescent process allows for simultaneous multiple mergers of ancestral lines.
Construction Of Markovian Coalescents
 Ann. Inst. Henri Poincar'e
, 1997
"... Partitionvalued and measurevalued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of masses with sum m, and whose evolution is determined by the following intuitive prescription: each pair of ma ..."
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Cited by 49 (16 self)
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Partitionvalued and measurevalued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of masses with sum m, and whose evolution is determined by the following intuitive prescription: each pair of masses of magnitudes x and y runs the risk of a binary collision to form a single mass of magnitude x+y at rate (x; y), for some nonnegative, symmetric collision rate kernel (x; y). Such processes with finitely many masses have been used to model polymerization, coagulation, condensation, and the evolution of galactic clusters by gravitational attraction. With a suitable choice of state space, and under appropriate restrictions on and the initial distribution of mass, it is shown that such processes can be constructed as Feller or Fellerlike processes. A number of further results are obtained for the additive coalescent with collision kernel (x; y) = x + y. This process, which arises fro...
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 48 (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.
Betacoalescents and continuous stable random trees
, 2006
"... Coalescents with multiple collisions, also known as Λcoalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case t ..."
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Cited by 47 (15 self)
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Coalescents with multiple collisions, also known as Λcoalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Λ is the Beta(2 − α, α) distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here we use a recent result of Birkner et al. to prove that Betacoalescents can be embedded in continuous stable random trees, about which much is known due to recent progress of Duquesne and Le Gall. Our proof is based on a construction of the DonnellyKurtz lookdown process using continuous random trees which is of independent interest. This produces a number of results concerning the smalltime behavior of Betacoalescents. Most notably, we recover an almost sure limit theorem of the authors for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and allele frequency spectrum associated with mutations in the context of population genetics.
On the Structure of QuasiStationary Competing Particles Systems
, 2007
"... We study point processes on the real line whose configurations X are locally finite, have a maximum, and evolve through increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q = {qij}i,j∈N. A probability measure on th ..."
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Cited by 41 (4 self)
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We study point processes on the real line whose configurations X are locally finite, have a maximum, and evolve through increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q = {qij}i,j∈N. A probability measure on the pair (X, Q) is said to be quasistationary if the joint law of the gaps of X and of Q is invariant under the evolution. A known class of universally quasistationary processes is given by the Ruelle Probability Cascades (RPC), which are based on hierarchally nested PoissonDirichlet processes. It was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasistationary. The main result of this work is a proof of this conjecture for the case where qij assume only a finite number of values. The result is of relevance for meanfield spin glass models, where the evolution corresponds to the cavity dynamics, and where the hierarchal organization of the Gibbs measure was first proposed as an ansatz.
Characterization of invariant measures at the leading edge for competing particle systems Ann. Probab. 33 (2005), 82–113 (Francis Comets) Université Paris Diderot  Paris 7, Mathématiques, case 7012, F75 205 Paris Cedex 13, France Email address: comets@
"... We study systems of particles on a line which have a maximum, are locally finite, and evolve with independent increments. ‘Quasistationary states ’ are defined as probability measures, on the σ algebra generated by the gap variables, for which the joint distribution of the gaps is invariant under t ..."
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Cited by 39 (2 self)
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We study systems of particles on a line which have a maximum, are locally finite, and evolve with independent increments. ‘Quasistationary states ’ are defined as probability measures, on the σ algebra generated by the gap variables, for which the joint distribution of the gaps is invariant under the time evolution. Examples are provided by Poisson processes with densities of the form, ρ(dx) = e−sx s dx, with s> 0, and linear superpositions of such measures. We show that conversely: any quasistationary state for the independent dynamics, with an exponentially bounded integrated density of particles, corresponds to a superposition of the above described probability measures, restricted to the relevant σalgebra. Among the systems for which this question is of some relevance are spinglass models of statistical mechanics, where the point process represents the collection of the free energies of distinct “pure states”, the time evolution corresponds to the addition of a spin variable, and the Poisson measures described above correspond to the socalled REM states. 2 A. RUZMAIKINA and M. AIZENMAN 1.
Random recursive trees and the BolthausenSznitman coalescent
 Electron. J. Probab
, 2005
"... We describe a representation of the BolthausenSznitman 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 ..."
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Cited by 38 (2 self)
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We describe a representation of the BolthausenSznitman 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 → ∞, and obtain a scaling law for the sizes of these blocks. We also consider the discretetime Markov chain giving the number of blocks after each collision of the coalescent restricted to [n]; we show that the transition probabilities of the timereversal of this Markov chain have limits as n → ∞. These results can be interpreted as describing a “postgelation ” phase of the BolthausenSznitman coalescent, in which a giant cluster containing almost all of the mass has already formed and the remaining small blocks are being absorbed. 1