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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.
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.
Journal of Theoretical Biology] (]]]])]]]–]]] Contents lists available at ScienceDirect Journal of Theoretical Biology
"... journal homepage: www.elsevier.com/locate/yjtbi How mutation affects evolutionary games on graphs ..."
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journal homepage: www.elsevier.com/locate/yjtbi How mutation affects evolutionary games on graphs
Submitted to the Bernoulli arXiv: arXiv:0000.0000 Interacting Particle Systems as Stochastic Social Dynamics
"... 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.
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"... Stable continuousstate branching processes with immigration and BetaFlemingViot processes with immigration ∗ ..."
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Stable continuousstate branching processes with immigration and BetaFlemingViot processes with immigration ∗
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
The ΛFlemingViot process and a connection with WrightFisher diffusion.
"... The ddimensional ΛFlemingViot generator acting on functions g(x), with x being a vector of d allele frequencies, can be written as a WrightFisher generator acting on functions g with a modified random linear argument of x induced by partitioning occurring in the ΛFlemingViot process. The eigen ..."
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The ddimensional ΛFlemingViot generator acting on functions g(x), with x being a vector of d allele frequencies, can be written as a WrightFisher generator acting on functions g with a modified random linear argument of x induced by partitioning occurring in the ΛFlemingViot process. The eigenvalues and right polynomial eigenvectors are easy to see from the representation. The twodimensional process, which has a onedimensional generator, is considered in detail. A nonlinear equation is found for the Green’s function. In a model with genic selection the fixation probability can be computed from an algorithm in the paper. An application in the infinitelymanyalleles ΛFlemingViot process is finding an interesting identity for the frequency spectrum of alleles that is based on sizebiassing. The moment dual process in the FlemingViot process is the usual Λcoalescent tree back in time. In the WrightFisher representation using a different set of polynomials gn(x) as test functions produces a dual death process which has a similarity to the Kingman coalescent and decreases by units of one. The eigenvalues of the process are analogous to the Jacobi polynomials when expressed in terms of gn(x), playing the role of xn.