The genealogy of Galton-Watson trees

Abstract

Take a continuous-time Galton-Watson tree and pick k distinct particles uniformly from those alive at a time T . What does their genealogical tree look like? The case k = 2 has been studied by several authors, and the near-critical asymptotics for general k appear in Harris, Johnston and Roberts (2018) [9]. Here we give the full picture. IntroductionLet L be a random variable taking values in {0, 1, 2, . . .}. Consider a continuous-time Galton-Watson tree starting with one initial particle, branching at rate 1, and with offspring distributed like L. Let N t be the number of particles alive at time t, and write f (s) := E[s L ] and F t (s) := E[s Nt ] for the generating functions associated with the process.Let T > 0, and on the event {N T ≥ k} pick k distinct particles U 1 , . . . , U k uniformly from those alive at time T . For each earlier time t ∈ [0, T ], define the equivalence relation ∼ t on {1, . . . , k} by i ∼ t j ⇐⇒ U i and U j share a common ancestor alive at time t.We let π k,L,T t denote the random partition of {1, . . . , k} corresponding to this equivalence relation. The process (π k,L,T t ) t∈[0,T ] , defined on the event {N T ≥ k}, is a right-continuous partition-valued stochastic process characterising the entire genealogical tree of U 1 , . . . , U k .Our goal is to describe the law of (π k,L,T t ) t∈[0,T ] conditioned on the event {N T ≥ k}, with a view towards the asymptotic regime T → ∞. We find that as T → ∞, there are marked differences in the qualitative behaviour of (π k,L,T t ) t∈[0,T ] depending on the mean number of offspring m := f ′ (1).Before we state our results in full generality in Section 3, we give an impression of the structure we expect to encounter by exploring the special case k = 2, which features as the focus of a chapter in the recent book [3], and on which the majority of the related literature concentrates. The case k = 2The case k = 2 amounts to choosing two particles uniformly from those alive at a time T from a tree with offspring distributed like L, and studying the time τ L,T in [0, T ] at which they last shared a common ancestor. In terms of the partition process (π 2,L,T t ) t∈[0,T ] , τ L,T is the time at which the single block {{1, 2}} splits into the pair of singletons {{1}, {2}}.The following characterisation of the law of τ L,T (which we will generalise later) was first given by Lambert [14].

The coalescent tree of a Markov branching process with generalised logistic growth

“…We consider the limit as the sampling time and carrying capacity simultaneously tend to infinity. We prove that the coalescent tree of a generalised logistic branching process coincides, in the limit, with the coalescent tree of a density-independent supercritical branching process, whose structure was recently determined [15]. Our result can be viewed as a universality or robustness result for the coalescent tree of a supercritical branching process.…”

“…Johnston [15] considered a continuous-time Galton-Watson branching process, determining the coalescent tree for finite and infinite sampling times, for the subcritical, critical, and supercritical regimes. We are especially interested in the supercritical regime.…”

“…Theorem 3.1 (Theorem 3.5 of [15]). Consider Example 2.5, which is a density-independent supercritical branching process.…”

“…Branching processes form a class of population models whose coalescent trees have been extensively studied [6,7,8,9,10,11,12,13,14]. In particular, we note that Johnston [15] recently gave a broad account of the coalescent tree arising from a continuous-time Galton-Watson branching process, including a limit result for the supercritical regime -which is our primary interest here.…”

Some properties of stationary continuous state branching processes

“…[21]). Other models have also been considered where the distribution of the genealogical tree or a sample of the current population can be explicitely described: linear birth-death process [34], continuous time Galton-Watson trees [22,25], Brownian tree [2] see also [1], splitting trees [31]. Some recent results on the coalescent process associated with some branching process by time-reversal can be found in [41,26,18].…”