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Register a new userThe Nested Kingman Coalescent: Speed of Coming Down from Infinity
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The nested Kingman coalescent describes the ancestral tree of a population undergoing neutral evolution at the level of individuals and at the level of species, simultaneously. We study the speed at which the number of lineages descends from infinity in this hierarchical coalescent process and prove the existence of an early-time phase during which the number of lineages at time $t$ decays as $ 2gamma/ct^2$, where $c$ is the ratio of the coalescence rates at the individual and species levels, and the constant $gammaapprox 3.45$ is derived from a recursive distributional equation for the number of lineages contained within a species at a typical time.
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Consider a population of fixed size that evolves over time. At each time, the genealogical structure of the population can be described by a coalescent tree whose branches are traced back to the most recent common ancestor of the population. As time goes forward, the genealogy of the population evolves, leading to what is known as an evolving coalescent. We will study the evolving coalescent for populations whose genealogy can be described by the Bolthausen-Sznitman coalescent. We obtain the limiting behavior of the evolution of the time back to the most recent common ancestor and the total length of the branches in the tree. By similar methods, we also obtain a new result concerning the number of blocks in the Bolthausen-Sznitman coalescent.
Let $mathbb{T}^d_N$, $dge 2$, be the discrete $d$-dimensional torus with $N^d$ points. Place a particle at each site of $mathbb{T}^d_N$ and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two particles meet, they coalesce into one. Denote by $C_N$ the first time the set of particles is reduced to a singleton. Cox [6] proved the existence of a time-scale $theta_N$ for which $C_N/theta_N$ converges to the sum of independent exponential random variables. Denote by $Z^N_t$ the total number of particles at time $t$. We prove that the sequence of Markov chains $(Z^N_{ttheta_N})_{tge 0}$ converges to the total number of partitions in Kingmans coalescent.
Full likelihood inference under Kingmans coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) method have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled approximations for accurate inference. Recently, more general $Lambda$- and $Xi$-coalescents have been observed to provide better modelling fits to some genetic data sets. We derive families of approximate CSDs for finite sites $Lambda$- and $Xi$-coalescents, and use them to obtain approximately optimal IS and PAC algorithms for $Lambda$-coalescents, yielding substantial gains in efficiency over existing methods.
This paper studies the spatial coalescent 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. These two situations are relevant if the coalescent is used to study the scaling limits for genealogies in Moran models respectively interacting Fisher-Wright diffusions (or Fleming-Viot processes), which is the key application of the present work. Our goal is to determine the longtime behavior with an initial population of countably many individuals per site restricted to a box $[-t^{alpha/2}, t^{alpha/2}]^2 cap Z^2$ and observed at time $t^beta$ with $1 geq beta geq alphage 0$. We study both asymptotics, as $ttoinfty$, for a fixed value of $alpha$ as the parameter $betain[alpha,1]$ varies, and for a fixed $beta$, as the parameter $alphain [0,beta]$ varies. This exhibits the genealogical structure of the mono-type clusters arising in 2-dimensional Moran and Fisher-Wright systems. (... for more see the actual preprint)
An important property of Kingmans coalescent is that, starting from a state with an infinite number of blocks, over any positive time horizon, it transitions into an almost surely finite number of blocks. This is known as `coming down from infinity. Moreover, of the many different (exchangeable) stochastic coalescent models, Kingmans coalescent is the `fastest to come down from infinity. In this article we study what happens when we counteract this `fastest coalescent with the action of an extreme form of fragmentation. We augment Kingmans coalescent, where any two blocks merge at rate $c>0$, with a fragmentation mechanism where each block fragments at constant rate, $lambda>0$, into its constituent elements. We prove that there exists a phase transition at $lambda=c/2$, between regimes where the resulting `fast fragmentation-coalescence process is able to come down from infinity or not. In the case that $lambda<c/2$ we develop an excursion theory for the fast fragmentation-coalescence process out of which a number of interesting quantities can be computed explicitly.
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