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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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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.
Interpreting Λcoalescent speed of coming down from infinity via particle representation of superprocesses
 In preparation
, 2008
"... Consider a Λcoalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number Nt of blocks at any positive time t> 0). We exhibit a deterministic function v: (0,∞) → (0,∞), such that Nt/v(t) → 1, almost ..."
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Cited by 27 (8 self)
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Consider a Λcoalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number Nt of blocks at any positive time t> 0). We exhibit a deterministic function v: (0,∞) → (0,∞), such that Nt/v(t) → 1, almost surely and in Lp for any p ≥ 1, as t → 0. Our approach relies on a novel martingale technique.
On the speed of coming down from infinity for Ξcoalescent processes
, 2010
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The spatial LambdaFlemingViot process: an eventbased construction and a lookdown representation
, 2013
"... We construct a measurevalued equivalent to the spatial ΛFlemingViot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of the sequence of reproduction events and obtain a quenched evolution of the local genetic diversities. To th ..."
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Cited by 4 (1 self)
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We construct a measurevalued equivalent to the spatial ΛFlemingViot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of the sequence of reproduction events and obtain a quenched evolution of the local genetic diversities. To this end, we use a particle representation which highlights the role of the genealogies in the attribution of types (or alleles) to the individuals of the population. This construction also enables us to clarify the statespace of the SLFV and to derive several path properties of the measurevalued process as well as of the labeled trees describing the genealogical relations between a sample of individuals. We complement it with a lookdown construction which provides a particle system whose empirical distribution at time t, seen as a process in t, has the law of the quenched SLFV. In all these results, the facts that we work with a fixed configuration of events and that reproduction occurs only locally in space introduce serious technical issues that are overcome by controlling the number of events occurring and of particles present in a given area over macroscopic time intervals.
Asymptotic sampling formulae for Λcoalescents
, 2012
"... We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a Λcoalescent. This allows us to derive an exa ..."
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Cited by 2 (0 self)
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We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a Λcoalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to ∞. Some of our results hold in the case of a general Λcoalescent that comes down from infinity, but we obtain more precise information under a regular variation assumption. In this case, we obtain results of independent interest for the time at which a mutation uniformly chosen at random was generated. This exhibits a phase transition at α = 3/2, where α ∈ (1, 2) is the exponent of regular variation. AMS 2000 Subject Classification. 60J25, 60F99, 92D25 Key words and phrases. Λcoalescents, speed of coming down from infinity, exchangeable