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47
The total external branch length of beta coalescents. Preprint available on http://arxiv.org/abs/1212.6070
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Interacting particle systems as stochastic social dynamics, Bernoulli (to appear
, 2013
"... The style of mathematical models known to probabilists as Interacting Particle Systems and exemplified by the Voter, Exclusion and Contact processes have found use in many academic disciplines. In many such disciplines the underlying conceptual picture is of a social network, where individuals meet ..."
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The style of mathematical models known to probabilists as Interacting Particle Systems and exemplified by the Voter, Exclusion and Contact processes have found use in many academic disciplines. In many such disciplines the underlying conceptual picture is of a social network, where individuals meet pairwise and update their “state ” (opinion, activity etc) in a way depending on the two previous states. This picture motivates a precise general setup we call Finite Markov Information Exchange (FMIE) processes. We briefly describe a few less familiar models (Averaging, Compulsive Gambler, Deference, Fashionista) suggested by the social network picture, as well as a few familiar ones.
Patterns of neutral diversity under general models of selective sweeps
"... Two major sources of stochasticity in the dynamics of neutral alleles result from resampling of finite populations (genetic drift) and the random genetic background of nearby selected alleles on which the neutral alleles are found (linked selection). There is now good evidence that linked selection ..."
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Two major sources of stochasticity in the dynamics of neutral alleles result from resampling of finite populations (genetic drift) and the random genetic background of nearby selected alleles on which the neutral alleles are found (linked selection). There is now good evidence that linked selection plays an important role in shaping polymorphism levels in a number of species. One of the best investigated models of linked selection is the recurrent full sweep model, in which newly arisen selected alleles fix rapidly. However, the bulk of selected alleles that sweep into the population may not be destined for rapid fixation. Here we develop a general model of recurrent selective sweeps in a coalescent framework, one that generalizes the recurrent full sweep model to the case where selected alleles do not sweep to fixation. We show that in a large population, only the initial rapid increase of a selected allele affects the genealogy at partially linked sites, which under fairly general assumptions are unaffected by the subsequent fate of the selected allele. We also apply the theory to a simple model to investigate the impact of recurrent partial sweeps on levels of neutral diversity, and find that for a given reduction in diversity, the impact of recurrent partial sweeps on the frequency spectrum at neutral sites is determined primarily by the frequencies achieved by the selected alleles. Consequently, recurrent sweeps of selected alleles to low frequencies can have a profound effect on levels of diversity but can leave the frequency spectrum relatively unperturbed. In fact, the limiting coalescent model under a high rate of sweeps to low frequency is identical to the standard neutral model. The general model of selective sweeps we describe goes some way towards providing a more flexible framework to describe genomic patterns of diversity than is currently available. 1
Generalized Fleming–Viot processes with immigration via stochastic flows of partitions
 ALEA Lat. Am. J. Probab. Math. Stat
"... Abstract. The generalized FlemingViot processes were defined in 1999 by Donnelly and Kurtz using a particle model and by Bertoin and Le Gall in 2003 using stochastic flows of bridges. In both methods, the key argument used to characterize these processes is the duality between these processes and e ..."
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Abstract. The generalized FlemingViot processes were defined in 1999 by Donnelly and Kurtz using a particle model and by Bertoin and Le Gall in 2003 using stochastic flows of bridges. In both methods, the key argument used to characterize these processes is the duality between these processes and exchangeable coalescents. A larger class of coalescent processes, called distinguished coalescents, was set up recently to incorporate an immigration phenomenon in the underlying population. The purpose of this article is to define and characterize a class of probability measurevalued processes called the generalized FlemingViot processes with immigration. We consider some stochastic flows of partitions of Z+, in the same spirit as Bertoin and Le Gall’s flows, replacing roughly speaking, composition of bridges by coagulation of partitions. Identifying at any time a population with the integers N: = {1,2,...}, the formalism of partitions is effective in the past as well as in the future especially when there are several simultaneous births. We show how a stochastic population may be directly embedded in the dual flow. An extra individual 0 will be viewed as an external generic immigrant ancestor, with a distinguished type, whose progeny represents the immigrants. The ”modified” lookdown construction of DonnellyKurtz is recovered when neither simultaneous multiple births nor immigration are taken into account. In the last part of the paper we give a sufficient criterion for the initial types extinction.
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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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.
On the speed of coming down from infinity for Ξcoalescent processes
, 2010
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On asympotics of the betacoalescents
 arXiv:1203.3110, 2012. 20 ROMAIN ABRAHAM AND JEANFRANÇOIS
"... We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree ..."
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We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen–Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a, b)coalescents with 0 < a < 1 leads to a simplified derivation of the known (2 − a)stable limit. We furthermore derive asymptotic expansions for the (centered) moments of the number of collisions and of the total branch length for the beta (1, b)coalescent by exploiting the method of sequential approximations. 1
A CONSTRUCTION OF A βCOALESCENT VIA THE PRUNING OF BINARY TREES
"... Abstract. Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β ( 3 1,)coalescent process. We also use the 2 2 continuous analogue of this construction, i.e. a pruning procedure on Aldous’s continuum random tree, to construct a co ..."
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Abstract. Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β ( 3 1,)coalescent process. We also use the 2 2 continuous analogue of this construction, i.e. a pruning procedure on Aldous’s continuum random tree, to construct a continuous state space process that has the same structure as the βcoalescent process up to some time change. These two constructions enable us to obtain results on the coalescent process such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event. hal00711518, version 2 9 Nov 2012 1.
ASYMPOTIC BEHAVIOR OF THE TOTAL LENGTH OF EXTERNAL BRANCHES FOR BETACOALESCENTS
"... Abstract. In this paper, we consider the Beta(2 − α,α)coalescents with 1 < α < 2 and study the moments of external branches, in particular the total external branch length L (n) ext of an initial sample of n individuals. For this class of coalescents, it has been proved that n α−1 (n) (d) T → ..."
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Abstract. In this paper, we consider the Beta(2 − α,α)coalescents with 1 < α < 2 and study the moments of external branches, in particular the total external branch length L (n) ext of an initial sample of n individuals. For this class of coalescents, it has been proved that n α−1 (n) (d) T → T, where T (n) is the length of an external branch chosen at random, and T is a known non negative random variable. We get the asymptotic behaviour of several moments of L (n) ext. As a consequence, we obtain that for Beta(2−α,α)coalescents with 1 < α < 2, lim n→+ ∞ n3α−5 E[(L (n) ext −n2−α E[T]) 2] =