The coalescent structure of Galton-Watson trees in varying environments

Abstract

We investigate the genealogy of a sample of $k\geq1$ particles chosen uniformly without replacement from a population alive at large times in a critical discrete-time Galton-Watson process in a varying environment (GWVE). We will show that subject to an explicit deterministic time-change involving only the mean and variances of the varying offspring distributions, the sample genealogy always converges to the same universal genealogical structure; it has the same tree topology as Kingman's coalescent, and the coalescent times of the $k-1$ pairwise mergers look like a mixture of independent identically distributed times. Our approach uses $k$ distinguished \emph{spine} particles and a suitable change of measure under which (a) the spines form a uniform sample without replacement, as required, but additionally (b) there is $k$-size biasing and discounting according to the population size. Our work significantly extends the spine techniques developed in Harris, Johnston, and Roberts \emph{[Annals Applied Probability, 2020]} for genealogies of uniform samples of size $k$ in near-critical continuous-time Galton-Watson processes, as well as a two-spine GWVE construction in Cardona and Palau \emph{[Bernoulli, 2021]}. Our results complement recent works by Kersting \emph{[Proc. Steklov Inst. Maths., 2022]} and Boenkost, Foutel-Rodier, and Schertzer \emph{[arXiv:2207.11612]}.

Figures (6)

• Figure 1: Example of the concatenation of $\textbf{t}$ (solid lines) and $\textbf{s}$ (dotted lines) at particle $u$ (red vertex). In the left hand-side picture, $u\notin \textbf{t}$. In the middle picture, $u\in \textbf{t}$ and $u$ is not a leaf. In the right hand-side picture, $u\in \textbf{t}$ and $u$ is a leaf.
• Figure 2: Example of a tree with 5 spines.
• Figure 3: Example of the spine concatenation of $(\textbf{t};\textbf{v}_1,\dots, \textbf{v}_5)$ and $(\textbf{s};\textbf{w}_1, \textbf{w}_2)$ at leaf $v_3$.
• Figure 4: Decomposition of the tree with spines $(\textbf{T};\textbf{V}_1,\textbf{V}_2,\textbf{V}_3,\textbf{V}_4)$ at generation $m=2$ into its restriction up to generation $2$ concatenated with the subtrees attached to particles $u, u_1,u_2,u_3$ and $u_4$. Particles without marks have dotted lines. The spines are the solid red lines.
• Figure 5: The action $\mathcal{O}$ applied to $(\mathbf{T};\mathbf{V}_1,\dots,\mathbf{V}_7)$, squeezes each line in the spines subtree to obtain a tree in $\mathcal{B}^7$.

Theorems & Definitions (33)

• Theorem 1.1
• Proposition 1
• proof
• Proposition 2
• proof : Proof of Proposition \ref{['prop:measure Q']}
• Proposition 3: Markov property for GWTVE with spines under $\mathbf{Q}_n^{(e,k,\theta)}$
• proof
• Proposition 4: A $k$-spine construction under $\mathbf{Q}_n^{(e,k,\theta)}$
• Proposition 5
• Remark 1