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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 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.
TREEVALUED RESAMPLING DYNAMICS (MARTINGALE PROBLEMS AND APPLICATIONS)
, 2009
"... The measurevalued FlemingViot process is a diffusion which models the evolution of allele frequencies in a multitype population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the “individuals” in the population at a fixed time. The goal of the present pap ..."
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Cited by 20 (10 self)
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The measurevalued FlemingViot process is a diffusion which models the evolution of allele frequencies in a multitype population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the “individuals” in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. Ultrametric spaces extend the class of discrete trees with edge length by allowing behavior such as infinitesimal short edges. We encode genealogies of the population at fixed times as elements in the space of (isometry classes of) ultrametric measure spaces. The additional probability measure on the ultrametric space allows to sample from the population. We equip this state space by the Gromovweak topology and use wellposed martingale problems to construct treevalued resampling dynamics for both the finite population (treevalued
The spatial Λcoalescent
 Elec. J. probab
, 2006
"... This paper extends the notion of the Λcoalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial Λcoalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the Λcoalescents that come down from ..."
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Cited by 14 (4 self)
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This paper extends the notion of the Λcoalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial Λcoalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the Λcoalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study spacetime asymptotics of spatial Λcoalescents on large tori in d ≥ 3 dimensions. Our results generalize and strengthen those of Greven et al. (2005), who studied the spatial Kingman coalescent in this context. AMS 2000 Subject Classification. Primary 60J25, 60K35 Key words and phrases. coalescent, Λcoalescent, structured coalescent, limit theorems The Λcoalescent, sometimes also called the coalescent with multiple collisions, is a Markov process Π whose state space is the set of partitions of the positive integers. The standard Λcoalescent Π starts at the partition of the positive integers into singletons, and its
Global divergence of spatial coalescents
 In preparation
, 2008
"... A class of processes called spatial Λcoalescents was recently introduced by Limic and Sturm (2006). In these models particles perform independent random walks on some underlying graph G. In addition, particles on the same site merge randomly according to some given coalescing mechanism. The goal of ..."
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Cited by 7 (4 self)
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A class of processes called spatial Λcoalescents was recently introduced by Limic and Sturm (2006). In these models particles perform independent random walks on some underlying graph G. In addition, particles on the same site merge randomly according to some given coalescing mechanism. The goal of the current work is to obtain several asymptotic results for these processes. If G = Z d, and the coalescing mechanism is Kingman’s coalescent, then starting with N particles at the origin, the number of particles is of order (log ∗ N) d at any fixed time (where log ∗ is the inverse tower function). At sufficiently large times this number is of order (log ∗ N) d−2. Betacoalescents behave similarly, with log log N in place of log ∗ N. Moreover, it is shown that on any graph and for general Λcoalescent, starting with infinitely many particles at a single site, the total number of particles will remain infinite at all times, almost surely.
Coalescent processes arising in a study of diffusive clustering
, 2009
"... This paper studies spatial coalescents on Z 2. In our setting, the partition elements are located at the sites of Z 2 and undergo local delayed coalescence and migration. That is, pairs of partition elements located at the same site coalesce into one partition element after exponential waiting times ..."
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This paper studies spatial coalescents on Z 2. In our setting, the partition elements are located at the sites of Z 2 and undergo local delayed coalescence and migration. That is, pairs of partition elements located at the same site coalesce into one partition element after exponential waiting times. In addition, the partition elements perform independent random walks. The system starts in either locally finite configurations or in configurations containing countably many partition elements per site. Our goal is to determine the longtime behavior with an initial population of countably many individuals per site restricted to a box [−t α/2, t α/2] 2 ∩ Z 2 and observed at time t β with 1 ≥ β ≥ α ≥ 0. We study both asymptotics, as t → ∞, for a fixed value of α as the parameter β ∈ [α, 1] varies, and for a fixed β = 1, as the parameter α ∈ [0, 1] varies. A new random object, the socalled coalescent with rebirth, is constructed and shown to arise in the limit. For sake of completeness, and in view of future applications we introduce the spatial coalescent with rebirth and study its longtime asymptotics as well. The present paper is the basis for forthcoming work [18], where the genealogies in interacting Moran
On spatial coalescents with multiple mergers in two dimensions
"... We consider the genealogy of a sample of individuals taken from a spatially structured population when the variance of the offspring distribution is relatively large. The space is structured into discrete sites of a graph G. If the population size at each site is large, spatial coalescents with mul ..."
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Cited by 1 (0 self)
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We consider the genealogy of a sample of individuals taken from a spatially structured population when the variance of the offspring distribution is relatively large. The space is structured into discrete sites of a graph G. If the population size at each site is large, spatial coalescents with multiple mergers, so called spatial Λcoalescents, for which ancestral lines migrate in space and coalesce according to some Λcoalescent mechanism, are shown to be appropriate approximations to the genealogy of a sample of individuals. We then consider as the graph G the two dimensional torus with side length 2L + 1 and show that as L tends to infinity, and time is rescaled appropriately, the partition structure of spatial Λcoalescents of individuals sampled far enough apart converges to the partition structure of a nonspatial Kingman coalescent. From a biological point of view this means that in certain circumstances both the spatial structure as well as larger variances of the underlying offspring distribution are harder to detect from the sample. However, supplemental simulations show that for moderately large L the different structure is still evident.
Genealogical constructions of population models
, 2014
"... Representations of population models in terms of countable systems of particles are constructed, in which each particle has a ‘type’, typically recording both spatial position and genetic type, and a level. For finite intensity models, the levels are distributed on [0, λ], whereas in the infinite i ..."
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Representations of population models in terms of countable systems of particles are constructed, in which each particle has a ‘type’, typically recording both spatial position and genetic type, and a level. For finite intensity models, the levels are distributed on [0, λ], whereas in the infinite intensity limit, at each time t, the joint distribution of types and levels is conditionally Poisson, with mean measure Ξ(t) × l where l denotes Lebesgue measure and Ξ(t) is a measurevalued population process. Key forces of ecology and genetics can be captured within this common framework. Models covered incorporate both individual and event based births and deaths, oneforone replacement, immigration, independent ‘thinning ’ and independent or exchangeable spatial motion and mutation of individuals. Since birth and death probabilities can depend