We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov's idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows - provided the sequence is tight - from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Lambda-coalescents. We show that the Lambda-coalescent defines an infinite (random) metric measure space if and only if the so-called "dust-free"-property holds.
In this article we state and prove a dichotomy, which is satisfied by the number of non-singleton blocks in Lambda-coalescents which have a dust component. The dichotomy is of a similar type to two well known results, also dichotomies in the behaviour of the Lambda-coalescent, of Pitman (1999) and Schweinsberg (2000). The three dichotomies combine together to give a complete picture of the qualitative behaviour of the singleton/non-singleton blocks of the Lambda-coalescent.
We study topological properties of random metric spaces which arise by Lambda-coalescents. These are stochastic processes, which start with an infinite number of lines and evolve through multiple mergers in an exchangeable setting. We show that the resulting Lambda-coalescent measure tree is compact iff the Lambda-coalescent comes down from infinity, i.e. only consists of finitely many lines at any positive time. If the Lambda-coalescent stays infinite, the resulting metric measure space is not even locally compact. Our results are based on general notions of compact and locally compact (isome…
We consider the lambda-coalescent processes with positive frequency of singleton clusters. The class in focus covers, for instance, the beta$(a,b)$-coalescents with $a>1$. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing L{é}vy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.
We construct an extension of the Lambda-coalescent to a spatial continuum and analyse its behaviour. Like the Lambda-coalescent, the individuals in our model can be separated into (i) a dust component and (ii) large blocks of coalesced individuals. We identify a five phase system, where our phases are defined according to changes in the qualitative behaviour of the dust and large blocks. We completely classify the phase behaviour, including necessary and sufficient conditions for the model to come down from infinity. We believe that two of our phases are new to Lambda-coalescent theory and dir…
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 Lambda-coalescent. 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 infinity. Some of our results hold in the case of a general Lambda-coalescent that comes down from infinity, but we obtain more precise information u…
The star-shaped $Λ$-coalescent and corresponding $Λ$-Fleming-Viot process where the $Λ$ measure has a single atom at unity are studied in this paper. The transition functions and stationary distribution of the $Λ$-Fleming-Viot process are derived in a two-type model with mutation. The distribution of the number of non-mutant lines back in time in the star-shaped $Λ$-coalescent is found. Extensions are made to a model with $d$ types, either with parent independent mutation or general Markov mutation, and an infinitely-many-types model when $d\to \infty$. An eigenfunction expansion for the trans…
We consider the genealogy of a sample of individuals taken from a spatially structured population when the variance of the offspring distribution is relatively large. The space is structured into discrete sites of a graph G. If the population size at each site is large, spatial coalescents with multiple mergers, so called spatial Lambda-coalescents, for which ancestral lines migrate in space and coalesce according to some Lambda-coalescent mechanism, are shown to be appropriate approximations to the genealogy of a sample of individuals. We then consider as the graph G the two dimensional torus…
We present and discuss new importance sampling schemes for the approximate computation of the sample probability of observed genetic types in the infinitely many sites model from population genetics. More specifically, we extend the 'classical framework', where genealogies are assumed to be governed by Kingman's coalescent, to the more general class of Lambda-coalescents and develop further Hobolth et. al.'s (2008) idea of deriving importance sampling schemes based on 'compressed genetrees'. The resulting schemes extend earlier work by Griffiths and Tavaré (1994), Stephens and Donnelly (2000),…
We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations in the partial tree. This is a first step to study the asymptotic distribution of a natural estimator of DNA mutation rate for species with large families.
Considering a random binary tree with $n$ labelled leaves, we use a pruning procedure on this tree in order to construct a $β(3/2,1/2)$-coalescent process. We also use the 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 $β$-coalescent process up to some time change. These two constructions unable 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 even…
Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and by Abraham and Delmas for the $β(3/2,1/2)$-coalescent. By considering a pruning procedure on stable Galton-Watson tree with $n$ labeled leaves, we give a representation of the discrete $β(1+α,1-α)$-coalescent, with $α\in [1/2,1)$ starting from the trivial partition of the $n$ first integers. The construction can also be made directly on the stable continuum L{é}vy tree, with parameter $1/α$, simultaneously for all $n$. This representation allows to use …
We study tree lengths in $Λ$-coalescents without a dust component from a sample of $n$ individuals. For the total length of all branches and the total length of all external branches we present laws of large numbers in full generality. The other results treat regularly varying coalescents with exponent 1, which cover the Bolthausen-Sznitman coalescent. The theorems contain laws of large numbers for the total length of all internal branches and of internal branches of order $a$ (i.e. branches carrying $a$ individuals out of the sample). These results transform immediately to sampling formulas i…
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…
A well-established model for the genealogy of a large population in equilibrium is Kingman's coalescent. For the population together with its genealogy evolving in time, this gives rise to a time-stationary tree-valued process. We study the sum of the branch lengths, briefly denoted as tree length, and prove that the (suitably compensated) sequence of tree length processes converges, as the population size tends to infinity, to a limit process with cadlag paths, infinite infinitesimal variance, and a Gumbel distribution as its equilibrium.
The goal of this paper is to study the lookdown model with selection in the case of a population containing two types of individuals, with a reproduction model which is dual to the $Λ$-coalescent. In particular we formulate the infinite population "$Λ$-lookdown model with selection". When the measure $Λ$ gives no mass to 0, we show that the proportion of one of the two types converges, as the population size $N$ tends to infinity, towards the solution of a stochastic differential equation driven by a Poisson point process. We show that one of the two types fixates in finite time if and only if…
We consider the compact space of pairs of nested partitions of $\mathbb N$, where by analogy with models used in molecular evolution, we call "gene partition" the finer partition and "species partition" the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of $Λ$-coalescents to nested partitio…
Formulas are provided for the cumulants and the moments of the time $T$ back to the most recent common ancestor of the Kingman coalescent. It is shown that both the $j$th cumulant and the $j$th moment of $T$ are linear combinations of the values $ζ(2m)$, $m\in\{0,\ldots,\lfloor j/2\rfloor\}$, of the Riemann zeta function $ζ$ with integer coefficients. The proof is based on a solution of a two-dimensional recursion with countably many initial values. A closely related strong convergence result for the tree length $L_n$ of the Kingman coalescent restricted to a sample of size $n$ is derived. The…
Take a continuous-time Galton-Watson tree. If the system survives until a large time $T$, then choose $k$ particles uniformly from those alive. What does the ancestral tree drawn out by these $k$ particles look like? Some special cases are known but we give a more complete answer. We concentrate on near-critical cases where the mean number of offspring is $1+μ/T$ for some $μ\in\mathbb{R}$, and show that a scaling limit exists as $T\to\infty$. Viewed backwards in time, the resulting coalescent process is topologically equivalent to Kingman's coalescent, but the times of coalescence have an inte…
The standard coalescent is widely used in evolutionary biology and population genetics to model the ancestral history of a sample of molecular sequences as a rooted and ranked binary tree. In this paper, we present a representation of the space of ranked trees as a space of constrained ordered matched pairs. We use this representation to define ergodic Markov chains on labeled and unlabeled ranked tree shapes analogously to transposition chains on the space of permutations. We show that an adjacent-swap chain on labeled and unlabeled ranked tree shapes has mixing time at least of order $n^3$, …
We define a multi-type coalescent point process of a general branching process with finitely many types. This multi-type coalescent fully describes the genealogy of the (quasi-stationary) standing population, providing types along ancestral lineages of all individuals in the standing population. We show that this point-process is a functional of a certain Markov chain defined by the planar embedding of the branching process. This Markov chain is further used to determine statistical properties of the ancestral tree, such as the time to the most recent common ancestor (MRCA) of two consecutive …
We give a de Finetti type representation for exchangeable random coalescent trees (formally described as semi-ultrametrics) in terms of sampling iid sequences from marked metric measure spaces. We apply this representation to define versions of tree-valued Fleming-Viot processes from a $Ξ$-lookdown model. As state spaces for these processes, we use, besides the space of isomorphy classes of metric measure spaces, also the space of isomorphy classes of marked metric measure spaces and a space of distance matrix distributions. This allows to include the case with dust in which the genealogical t…
We design a theoretic tree-based functional representation of a class of Feynman-Kac particle distributions, including an extension of the Wick product formula to interacting particle systems. These weak expansions rely on an original combinatorial, and permutation group analysis of a special class of forests. They provide refined non asymptotic propagation of chaos type properties, as well as sharp $\LL\_p$-mean error bounds, and laws of large numbers for $U$-statistics. Applications to particle interpretations of the top eigenvalues, and the ground states of Schrödinger semigroups are also d…
In mathematical population genetics, it is well known that one can represent the genealogy of a population by a tree, which indicates how the ancestral lines of individuals in the population coalesce as they are traced back in time. As the population evolves over time, the tree that represents the genealogy of the population also changes, leading to a tree-valued stochastic process known as the evolving coalescent. Here we will consider the evolving coalescent for populations whose genealogy can be described by a beta coalescent, which is known to give the genealogy of populations with very la…
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 le…
We derive the asymptotic distribution of the total length $L_n$ of a $\operatorname {Beta}(2-α,α)$-coalescent tree for $1<α<2$, starting from $n$ individuals. There are two regimes: If $α\le1/2(1+\sqrt{5})$, then $L_n$ suitably rescaled has a stable limit distribution of index $α$. Otherwise $L_n$ just has to be shifted by a constant (depending on $n$) to get convergence to a nondegenerate limit distribution. As a consequence, we obtain the limit distribution of the number $S_n$ of segregation sites. These are points (mutations), which are placed on the tree's branches according to a Poisson p…
We provide scaling limits for the block counting process and the fixation line of $Λ$-coalescents as the initial state $n$ tends to infinity under the assumption that the measure $Λ$ on $[0,1]$ satisfies $\int_{[0,1]}u^{-1}(Λ-bλ)({\rm d}u)<\infty$ for some $b>0$. Here $λ$ denotes the Lebesgue measure. The main result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein--Uhlenbeck type process as $n$ tends to infinity. The result is applied to beta coalescents with parameters $1$ and $b>0$. We split the generators into two parts…
We investigate a new model for populations evolving in a spatial continuum. This model can be thought of as a spatial version of the Lambda-Fleming-Viot process. It explicitly incorporates both small scale reproduction events and large scale extinction-recolonisation events. The lineages ancestral to a sample from a population evolving according to this model can be described in terms of a spatial version of the Lambda-coalescent. Using a technique of Evans(1997), we prove existence and uniqueness in law for the model. We then investigate the asymptotic behaviour of the genealogy of a finite n…
The evolving Kingman coalescent is the tree-valued process which records the time evolution undergone by the genealogies of Moran populations. We consider the associated process of total external tree length of the evolving Kingman coalescent and its asymptotic behaviour when the number of leaves of the tree tends to infinity. We show that on the time-scale of the Moran model slowed down by a factor equal to the population size, the (centred and rescaled) external length process converges to a stationary Gaussian process with almost surely continuous paths and covariance function $c(s,t)=\Big(…
In these expository notes, we describe some features of the multiplicative coalescent and its connection with random graphs and minimum spanning trees. We use Pitman's proof of Cayley's formula, which proceeds via a calculation of the partition function of the additive coalescent, as motivation and as a launchpad. We define a random variable which may reasonably be called the empirical partition function of the multiplicative coalescent, and show that its typical value is exponentially smaller than its expected value. Our arguments lead us to an analysis of the susceptibility of the Erdős-Rény…
We consider a class of density-dependent branching processes which generalises exponential, logistic and Gompertz growth. A population begins with a single individual, grows exponentially initially, and then growth may slow down as the population size moves towards a carrying capacity. At a time while the population is still growing superlinearly, a fixed number of individuals are sampled and their coalescent tree is drawn. Taking the sampling time and carrying capacity simultaneously to infinity, we prove convergence of the coalescent tree to a limiting tree which is in a sense universal over…
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…
We define a doubly infinite, monotone labeling of Bienayme-Galton-Watson (BGW) genealogies. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_i; i\ge 1)$, where $A_i$ is the coalescence time between individuals i and i+1. There is a Markov process of point measures $(B_i; i\ge 1)$ keeping track of more ancestral relationships, such that $A_i$ is also the first point mass of $B_i$. This process of point measures is also closely related to an inhomogeneous spine decomposition of the lineage of the first surviving particle in gene…
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 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.
Kingman's coalescent is a random tree that arises from classical population genetic models such as the Moran model. The individuals alive in these models correspond to the leaves in the tree and the following two laws of large numbers concerning the structure of the tree-top are well-known: (i) The (shortest) distance, denoted by $T_n$, from the tree-top to the level when there are $n$ lines in the tree satisfies $nT_n \xrightarrow{n\to\infty} 2$ almost surely; (ii) At time $T_n$, the population is naturally partitioned in exactly $n$ families where individuals belong to the same family if the…
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…
The paper establishes a weak version of Horton self-similarity for a tree representation of Kingman's coalescent process. The proof is based on a Smoluchowski-type system of ordinary differential equations for the number of branches of a given Horton-Strahler order in a tree that represents Kingman's N-coalescent process with a constant kernel, in a hydrodynamic limit. We also demonstrate a close connection between the combinatorial Kingman's tree and the combinatorial level set tree of a white noise, which implies Horton self-similarity for the latter.
We consider Beta$(2-α, α)$-coalescents with parameter range $1 <α<2$ starting from $n$ leaves. The length $\ell^{(n)}_r$ of order $r$ in the $n$-Beta$(2-α, α)$-coalescent tree is defined as the sum of the lengths of all branches that carry a subtree with $r$ leaves. We show that for any $s \in \mathbb N$ the vector of suitably centered and rescaled lengths of orders $1\le r \le s$ converges in distribution to a multivariate stable distribution as the number of leaves tends to infinity.
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 $ 2γ/ct^2$, where $c$ is the ratio of the coalescence rates at the individual and species levels, and the constant $γ\approx 3.45$ is derived from a recursive distributional equation for the number of lin…
We introduce a colored coalescent process which recovers random colored genealogical trees. Here a colored genealogical tree has its vertices colored black or white. Moving backward along the colored genealogical tree, the color of vertices may change only when two vertice coalesce. The rule that governs the change of color involves a parameter $x$. When $x=1/2$, the colored coalescent process can be derived from a variant of the Wright-Fisher model for a haploid population in population genetics. Explicit computations of the expectation and the cumulative distribution function of the coalesce…
Coalescents with multiple collisions, also known as $Λ$-coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure $Λ$ is the $\operatorname {Beta}(2-α,α)$ distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here, we use a recent result of Birkner et al. to prove that Beta-coalescents can be embedd…
Incomplete lineage sorting (ILS) is a common source of gene tree incongruence in multilocus analyses. A large number of methods have been developed to infer species trees in the presence of ILS. Here we provide a mathematical analysis of several coalescent-based methods. Our analysis is performed on a three-taxon species tree and assumes that the gene trees are correctly reconstructed along with their branch lengths.
The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths), culminating in the $β$-family of Aldous' branching models. We next introduce real trees (trees as metric spaces) and show how to study them through their contour, provided they are properly measured and ordered. We then focus on the reduced tree, or coalescent tree, which is the tree spanned by individuals/species alive at the same fixed time. We show how red…
We dedicate this paper to Sir John Kingman on his 70th Birthday. In modern mathematical population genetics the ancestral history of a population of genes back in time is described by John Kingman's coalescent tree. Classical and modern approaches model gene frequencies by diffusion processes. This paper, which is partly a review, discusses how coalescent processes are dual to diffusion processes in an analytic and probabilistic sense. Bochner (1954) and Gasper (1972) were interested in characterizations of processes with Beta stationary distributions and Jacobi polynomial eigenfunctions. We d…
For a finite measure $\varLambda$ on $[0,1]$, the $\varLambda$-coalescent is a coalescent process such that, whenever there are $b$ clusters, each $k$-tuple of clusters merges into one at rate $\int_0^1x^{k-2}(1-x)^{b-k}\varLambda(\mathrm{d}x)$. It has recently been shown that if $1<α<2$, the $\varLambda$-coalescent in which $\varLambda$ is the $\operatorname {Beta}(2-α,α)$ distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an $α$-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptoti…
We consider the reconstruction of a phylogeny from multiple genes under the multispecies coalescent. We establish a connection with the sparse signal detection problem, where one seeks to distinguish between a distribution and a mixture of the distribution and a sparse signal. Using this connection, we derive an information-theoretic trade-off between the number of genes, $m$, needed for an accurate reconstruction and the sequence length, $k$, of the genes. Specifically, we show that to detect a branch of length $f$, one needs $m = Θ(1/[f^{2} \sqrt{k}])$.
Kingman (1978)'s representation theorem states that any exchangeable partition of $\mathbb{N}$ can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e. ordered partition of $\mathbb{N}$) can be represented as a paintbox based on an interval-partition (Gnedin 1997). Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a paintbox based on a random non-decreasing process valued in interval-partitions, called nested interval-partition, generalizing the notion of comb metric space …
Rice University（莱斯大学）；Massachusetts Institute of Technology（麻省理工学院）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
The reconstruction of a species phylogeny from genomic data faces two significant hurdles: 1) the trees describing the evolution of each individual gene--i.e., the gene trees--may differ from the species phylogeny and 2) the molecular sequences corresponding to each gene often provide limited information about the gene trees themselves. In this paper we consider an approach to species tree reconstruction that addresses both these hurdles. Specifically, we propose an algorithm for phylogeny reconstruction under the multispecies coalescent model with a standard model of site substitution. The mu…
We consider the problem of estimating species trees from unrooted gene tree topologies in the presence of incomplete lineage sorting, a common phenomenon that creates gene tree heterogeneity in multilocus datasets. One popular class of reconstruction methods in this setting is based on internode distances, i.e. the average graph distance between pairs of species across gene trees. While statistical consistency in the limit of large numbers of loci has been established in some cases, little is known about the sample complexity of such methods. Here we make progress on this question by deriving …
One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array of random graph models with degree exponent $τ\in (3,4)$, distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{(τ-3)/(τ-1)}$. In this paper we study the metric space structure of maximal components of the multiplicative coalescent, i…
