The size of the last merger and time reversal in $Λ$-coalescents

Lifting linear preferential attachment trees yields the arcsine coalescent

We consider linear preferential attachment trees which are specific scale-free trees also known as (random) plane-oriented recursive trees. Starting with a linear preferential attachment tree of size $n$ we show that repeatedly applying a so-called lifting yields a continuous-time Markov chain on linear preferential attachment trees. Each such tree induces a partition of $\{1, \ldots, n\}$ by placing labels in the same block if and only if they are attached to the same node in the tree. Our main result is that this Markov chain on linear preferential attachment trees induces a partition valued…

Genealogies and inference for populations with highly skewed offspring distributions

We review recent progress in the understanding of the role of multiple- and simultaneous multiple merger coalescents as models for the genealogy in idealised and real populations with exceptional reproductive behaviour. In particular, we discuss models with `skewed offspring distribution' (or under other non-classical evolutionary forces) which lead in the single locus haploid case to multiple merger coalescents, and in the multi-locus diploid case to simultaneous multiple merger coalescents. Further, we discuss inference methods under the infinitely-many sites model which allow both model sel…

复制子合并过程

The replicator coalescent

该研究提出复制子合并过程，一种多类型区块随机合并模型，揭示其从无穷多区块出发的瞬时动态与进化博弈论中复制子方程的显著联系，并分析其初始阶段行为及后续随机演化。

Universality classes for the coalescent structure of heavy-tailed Galton-Watson trees

Consider a population evolving as a critical continuous-time Galton-Watson (GW) tree. Conditional on the population surviving until a large time $T$, sample $k$ individuals uniformly at random (without replacement) from amongst those alive at time $T$. What is the genealogy of this sample of individuals? In cases where the offspring distribution has finite variance, the probabilistic properties of the joint ancestry of these $k$ particles are well understood, as seen in \cite{HJR20, J19}. In the present article, we study the joint ancestry of a sample of $k$ particles under the following regim…

Multiple-merger genealogies -- models, consequences, inference

Trees corresponding to $Λ$- and $Ξ$-$n$-coalescents can be both quite similar and fundamentally different compared to bifurcating tree models based on Kingman's $n$-coalescent. This has consequences for inference of a well-fitting gene genealogy as well as for assessing biological properties of species having such sample genealogies. Here, mathematical properties concerning clade sizes in the tree as well as changes of the tree when the samples are enlargened are highlighted. To be used as realistic genealogy models for real populations, an extension for changing population sizes is discussed.

Random recursive trees and the Bolthausen-Sznitman coalescent

We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the final collision converges as n tends to infinity, and obtain a scaling law for the sizes of these blocks. We also consider the discrete-time Markov chain giving the number of blocks after each collision of the coalescent restricted to [n]; we show that the transition probabilities of the time-revers…

根据抽奖式繁殖进化的单倍体群体中的基因谱系

Gene genealogies in haploid populations evolving according to sweepstakes reproduction

研究单倍体群体抽奖式繁殖中的基因谱系，构建群体遗传模型，无需强假设恢复新突变数，得出连续时间合并过程等结果，还发现不同条件下泛函近似情况及对种群祖先条件设定的影响。

The site frequency spectrum for general coalescents

General genealogical processes such as $Λ$- and $Ξ$-coalescents, which respectively model multiple and simultaneous mergers, have important applications in studying marine species, strong positive selection, recurrent selective sweeps, strong bottlenecks, large sample sizes, and so on. Recently, there has been significant progress in developing useful inference tools for such general models. In particular, inference methods based on the site frequency spectrum (SFS) have received noticeable attention. Here, we derive a new formula for the expected SFS for general $Λ$- and $Ξ$-coalescents, whic…

变化环境中 Galton-Watson 树的溯祖结构

The coalescent structure of Galton-Watson trees in varying environments

本文研究临界变化环境 Galton-Watson 过程的均匀样本谱系，利用 k 脊与测度变换证明经时间变换后收敛为 Kingman 型普适溯祖结构。

A new encoding of coalescent processes. Applications to the additive and multiplicative cases

We revisit the discrete additive and multiplicative coalescents, starting with $n$ particles with unit mass. These cases are known to be related to some "combinatorial coalescent processes": a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process $(G(n,p))_p$ in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as $n\to +\infty$. We propose to use the Prim order on the vertices instead of the…

Total length of the genealogical tree for quadratic stationary continuous-state branching processes

We prove the existence of the total length process for the genealogical tree of a population model with random size given by a quadratic stationary continuous-state branching processes. We also give, for the one-dimensional marginal, its Laplace transform as well as the fluctuation of the corresponding convergence. This result is to be compared with the one obtained by Pfaffelhuber and Wakolbinger for constant size population associated to the Kingma coalescent. We also give a time reversal property of the number of ancestors process at all time, and give a description of the so-called lineage…