The quenched coalescent for structured diploid populations with large migrations and uneven offspring distributions

In this work we describe a new model for the evolution of a diploid structured population backwards in time that allows for large migrations and uneven offspring distributions. The model generalizes both the mean-field model of Birkner et al. [\textit{Electron. J. Probab.} 23: 1-44 (2018)] and the haploid structured model of Möhle [\textit{Theor. Popul. Biol.} 2024 Apr:156:103-116]. We show convergence, with mild conditions on the joint distribution of offspring frequencies and migrations, of gene genealogies conditional on the pedigree to a time-inhomogeneous coalescent process driven by a Poisson point process $Ψ$ that records the timing and scale of large migrations and uneven offspring distributions. This quenched scaling limit demonstrates a significant difference in the predictions of the classical annealed theory of structured coalescent processes. In particular, the annealed and quenched scaling limits coincide if and only if these large migrations and uneven offspring distributions are absent. The proof proceeds by the method of moments and utilizes coupling techniques from the theory of random walks in random environments. Several examples are given and their quenched scaling limits established.

💡 Research Summary

This paper introduces a rigorous quenched coalescent framework for diploid structured populations that experience occasional large‑scale migrations and highly skewed offspring distributions. The authors model the population on a finite directed graph G = (V,E), where each vertex v represents a deme of size N(v)=⌊s(v)·N⌋. In each discrete generation a random migration vector m assigns a proportion m_e of individuals from deme v to deme w for every edge e=(v,w), while a random symmetric matrix V_v describes the number of offspring produced by each ordered pair of parents within deme v. The joint law of (V,m) is assumed exchangeable within demes, independent across generations, and satisfies natural feasibility constraints (no self‑migration, no self‑fertilisation, conservation of total population size after accounting for migration).

The central object of study is the genealogy of a sample of n genes conditional on the realized pedigree G_N, i.e., the quenched genealogy. The authors prove that, under mild integrability conditions and assuming that large migration events and extreme reproductive events are sufficiently rare, the quenched genealogy converges (as N→∞) to a time‑inhomogeneous structured coalescent χ_n driven by a Poisson point process Ψ. Between the atoms of Ψ, lineages evolve exactly as in the classical structured coalescent: each block moves independently from deme v to deme w at a “neutral” migration rate μ_{v,w} and any pair of blocks in the same deme coalesces at rate κ_v. At the atoms of Ψ, a paint‑box construction—generalising Kingman’s paint‑box—determines a simultaneous multiple‑merger event whose pattern is dictated by the ordered offspring frequencies eV recorded at that generation. In biological terms, the atoms correspond either to generations where a macroscopic fraction of the total population migrates in a single wave, or to generations dominated by a prodigious progenitor whose offspring constitute a non‑vanishing proportion of the next generation.

The proof proceeds via the method of moments. By a theorem of Feng, Niu and Wang (2025), it suffices to show convergence of all finite‑dimensional distributions of the quenched process. The authors construct l conditionally independent copies of the discrete‑time ancestral process on the same pedigree and prove that their joint moments converge to those of l independent copies of the limiting time‑inhomogeneous coalescent conditioned on Ψ. The argument hinges on a separation‑of‑scales principle: (i) large‑scale pedigree events are rare on the coalescent time‑scale but, when they occur, they affect all copies in essentially the same way; (ii) during the overwhelming majority of “small” generations, lineages experience neutral migration and coalescence, and their decisions become asymptotically independent across copies. Moment bounds and law‑of‑large‑numbers type estimates control the small‑scale behaviour, while a coupling argument (inspired by earlier work on quenched coalescents) handles the large‑scale jumps.

The paper also provides several concrete examples illustrating the breadth of the framework. A two‑deme Wright–Fisher model with neutral migration reproduces the setting of Wang, Birkner and Lenz (2017) and exhibits occasional whole‑population migration events captured by Ψ. A second two‑deme model mirrors the bottleneck setting of Döring, Fearnhead, Birkner and Wilke (2024), where only extreme reproductive events occur. A model converging to a β‑coalescent with β‑distributed migration sizes demonstrates how the intensity measure of Ψ can be tuned to produce heavy‑tailed jump behaviour. Finally, a discrete approximation of a spatial Ξ‑Fleming–Viot process on the two‑torus shows that the theory extends to continuous‑space settings with local migration kernels. In each case the authors verify the required integrability conditions and explicitly describe the limiting Poisson point process and paint‑box law.

A key conceptual contribution is the clear distinction between annealed and quenched limits. The authors prove that the annealed (averaged‑over‑pedigree) genealogy coincides with the quenched limit only when the Poisson point process Ψ has no atoms—that is, when large migrations and extreme offspring events are absent. Otherwise, the quenched genealogy exhibits additional time‑inhomogeneity and cross‑locus dependence that are invisible to annealed analyses. This result underscores the importance of accounting for the realized pedigree in structured populations, especially when rare but massive demographic events are biologically plausible.

The paper concludes with a discussion comparing its findings to previous simulation studies, outlining potential extensions (continuous‑time models, selection, inference frameworks), and suggesting that the quenched perspective may become essential for accurate demographic inference from genomic data in spatially structured species.