This paper extends the notion of the $\la$-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial $Λ$-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the $Λ$-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study space-time asymptotics of spatial $Λ$-coalescents on large tori in $d\ge 3$ dimensions. Our results generalize and strengthen those of Greven et al. (2005), who studied the spatial Kingman coalescent in this context.
In this paper we look at the asymptotic number of r-caterpillars for $Λ$-coalescents which come down from infinity, under a regularly varying assumption. An r-caterpillar is a functional of the coalescent process started from $n$ individuals which, roughly speaking, is a block of the coalescent at some time, formed by one line of descend to which r-1 singletons have merged one by one. We show that the number of $r$-caterpillars, suitably scaled, converge to an explicit constant as the sample size n goes to infinity.
The $\X$-coalescent processes were initially studied by Möhle and Sagitov (2001), and introduced by Schweinsberg (2000) in their full generality. They arise in the mathematical population genetics as the complete class of scaling limits for genealogies of Cannings' models. The $\X$-coalescents generalize $Λ$-coalescents, where now simultaneous multiple collisions of blocks are possible. The standard version starts with infinitely many blocks at time 0, and it is said to come down from infinity if its number of blocks becomes immediately finite, almost surely. This work builds on the technique …
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…
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…
Consider a $Λ$-coalescent 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 $N_t$ of blocks at any positive time $t>0$). We exhibit a deterministic function $v:(0,\infty)\to(0,\infty)$ such that $N_t/v(t)\to1$, almost surely, and in $L^p$ for any $p\geq1$, as $t\to0$. Our approach relies on a novel martingale technique.
We quantify the manner in which the beta coalescent $Π=\{ Π(t), t\geq 0\},$ with parameters $a\in (0, 1),$ $b>0,$ comes down from infinity. Approximating $Π$ by its restriction $Π^n$ to $[n]\:= \{1, \ldots, n\},$ the suitably rescaled block counting process $n^{-1}\#Π^n(tn^{a-1})$ has a deterministic limit, $c(t)$, as $n\to\infty.$ An explicit formula for $c(t)$ is provided in Theorem 1. The block size spectrum $(\mathfrak{c}_{1}Π^n(t), \ldots, \mathfrak{c}_{n}Π^n(t)),$ where $\mathfrak{c}_{i}Π^n(t)$ counts the number of blocks of size $i$ in $Π^n(t),$ captures more refined information about t…
An important property of Kingman's 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, Kingman's 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 Kingman's coalescent, where any two blocks merge at rate $…
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…
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 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 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…
The Ancestral Selection Graph (ASG) is an important genealogical process which extends the well-known Kingman coalescent to incorporate natural selection. We show that the number of lineages of the ASG with and without mutation is asymptotic to $2/t$ as $t\to 0$, in agreement with the limiting behaviour of the Kingman coalescent. We couple these processes on the same probability space using a Poisson random measure construction that allows us to precisely compare their hitting times. These comparisons enable us to characterise the speed of coming down from infinity of the ASG as well as its fl…
We study several fundamental properties of a class of stochastic processes called spatial Lambda-coalescents. In these models, a number of particles perform independent random walks on some underlying graph G. In addition, particles on the same vertex merge randomly according to a given coalescing mechanism. A remarkable property of mean-field coalescent processes is that they may come down from infinity, meaning that, starting with an infinite number of particles, only a finite number remains after any positive amount of time, almost surely. We show here however that, in the spatial setting, …
We describe a new general connection between $Λ$-coalescents and genealogies of continuous-state branching processes. This connection is based on the construction of an explicit coupling using a particle representation inspired by the lookdown process of Donnelly and Kurtz. This coupling has the property that the coalescent comes down from infinity if and only if the branching process becomes extinct, thereby answering a question of Bertoin and Le Gall. The coupling also offers new perspective on the speed of coming down from infinity and allows us to relate power-law behavior for $N^Λ(t)$ to …
For a class of $Λ$-Fleming-Viot processes with Brownian spatial motion in $\mathbb{R}^d$ whose associated $Λ$-coalescents come down from infinity, we obtain sharp global and local modulus of continuities for the ancestry processes recovered from the lookdown representations. As applications, we prove both global and local modulus of continuities for the $Λ$-Fleming-Viot support processes. In particular, if the $Λ$-coalescent is the Beta$(2-β,β)$ coalescent for $β\in(1,2]$ with $β=2$ corresponding to Kingman's coalescent, then for $h(t)=\sqrt{t\log (1/t)}$, the global modulus of continuity hold…
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…
We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a $Λ$-coalescent, and fragmentation dislocates at finite rate an individual block into sub-blocks of infinite size. We call these partition-valued processes, simple EFC processes, and study the question whether such a process, when started with infinitely many blocks, can visit partitions with a finite number of blocks or not. When this occurs, one says that the process comes down from infinity. We introduce two sharp parameters $θ_{\star}\leq θ^{\star}\in…
We approximate stochastic processes in finite dimension by dynamical systems. We provide trajectorial estimates which are uniform with respect to the initial condition for a well chosen distance. This relies on some non-expansivity property of the flow, which allows to deal with non-Lipschitz vector fields. We use the stochastic calculus and follow the martingale technics initiated in Berestycki and al [5] to control the fluctuations. Our main applications deal with the short time behavior of stochastic processes starting from large initial values. We state general properties on the coming dow…
We study a spatial asymptotic behaviour at infinity of kernels $p_t(x)$ for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits $$ \lim_{r \to \infty} \frac{p_t(rθ-y)}{t \, ν(rθ)}, \quad t \in T, \ \ θ\in E, \ \ y \in \mathbb R^d, $$ exist and can be effectively computed. Here $ν$ is the corresponding Lévy density, $T \subset (0,\infty)$ is a bounded time-set and $E$ is a subset of the unit sphere in $\mathbb R^d$, $d \geq 1$. Our results are local on the unit sphere. They apply to a wide class of convolution …
