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A conditional coalescent for diploid exchangeable population models given the pedigree

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Pith's one-line read Conditioning on a fixed diploid pedigree changes the limiting coalescent: multiple mergers occur only at pedigree-fixed times of large families, whereas the classical coalescent averages over them.

desk verdict Genuinely new quenched coalescent limit with a new limiting process, but the proof's central mixing assumption is explicitly heuristic, so the main theorem is not yet proven as written. read the letter →

arxiv 2505.15481 v1 pith:2T3GGQ6K submitted 2025-05-21 math.PR

Frederic Alberti , Matthias Birkner , Wai-Tong Louis Fan , John Wakeley This is my paper · ORCID
classification math.PR MSC 60J9092D1060K3760J95
keywords coalescenttheorypopulationpedigreequenchedlimitinhomogeneousmultiplemergerssite-frequencyspectrumdiploidexchangeablemodelgenerationswithlargeindividualprogeny
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Classical coalescent theory predicts genetic variation by averaging over all possible reproductive histories, which erases the population pedigree. This paper asks what happens instead when the pedigree—the complete record of which individuals were parents of whom—is fixed, and answers it for the standard diploid exchangeable bi-parental (Cannings) model in the large-population limit. The answer is a new object, the inhomogeneous $(\Psi,c_{\mathrm{pair}})$-coalescent: multiple mergers happen only at times fixed by generations with unusually large individual offspring numbers (GLIPs), recorded in a Poisson point process $\Psi$, while ordinary binary mergers happen at constant rate $c_{\mathrm{pair}}$. This pedigree-conditional (quenched) limit differs from the pedigree-averaged (annealed) $\Xi$-coalescent whenever the offspring distribution has mass at nonzero frequencies, so the paper concludes that a fixed pedigree leaves observable signatures in multi-locus statistics such as the site-frequency spectrum. The proof works for samples of arbitrary size and covers several concrete models, including two kinds of diploid Beta-coalescents.

What carries the argument

The central object is the inhomogeneous $(\Psi,c_{\mathrm{pair}})$-coalescent, a partition-valued process built from two ingredients: a Poisson point process $\Psi$ whose atoms $(t,x)$ trigger 'paintbox mergers' with probabilities $p(x;\xi,\eta)$ given in (15), and independent pair mergers at rate $c_{\mathrm{pair}}$. Because $\Psi$ may be dense in time, the construction uses a coagulator: each partition $\alpha$ defines an operator $\mathrm{Coag}_\alpha$ that merges blocks of the current partition according to the blocks of $\alpha$, and the coalescent is the chronologically ordered product of these operators over the atoms of the driving point process, giving a stochastic flow. The bridge from the finite population to this limit is the $\varepsilon$-naive coalescent driven by the cutoff point process $\Psi^{(N)}_\varepsilon$, built from offspring frequencies through $\widetilde V=(V_{(1)}/4N,V_{(1)}/4N,V_{(2)}/4N,\dots)$; Lemma 4.2 shows that the coarse-grained pedigree transition probabilities equal the paintbox probabilities $p(\widetilde V;\xi,\eta)$ up to $O(1/N)$. Coupling two conditionally independent copies of the coalescent on the same pedigree, interpreted as two unlinked loci, and applying a separation-of-timescales argument isolates the limiting jump rates $q_{\xi,\eta}$ that define the quenched process.

What would settle it

Simulate the occasional-large-family model (Section 7.4, $\gamma=1$) at increasing population sizes $N$, fix one realization of the pedigree, and estimate the probability that three sampled lineages undergo a simultaneous triple merger at rescaled times away from generations with large offspring numbers; Theorem 3.8 predicts all such multiple mergers occur exactly at GLIP times, so a nonzero triple-merger rate away from those times that does not vanish as $N\to\infty$ would refute the convergence.

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Extended reading notes

Core claim

The paper's central claim is Theorem 3.8: under the diploid Cannings model with no selfing, for every sample size $n$, every finite set of times $t_1,\dots,t_k$, and every sequence of partition states $\xi_1,\dots,\xi_k$, the pedigree-conditional probabilities $P^{(N)}(\Pi^{N,n}_{\lfloor t_1/c_N\rfloor}=\xi_1,\dots,\Pi^{N,n}_{\lfloor t_k/c_N\rfloor}=\xi_k \mid \mathcal{A}^{(N)})$ converge to $P(\Pi^n_{t_1}=\xi_1,\dots,\Pi^n_{t_k}=\xi_k\mid\Psi)$, where $\Pi^n$ is the inhomogeneous $(\Psi,c_{\mathrm{pair}})$-coalescent of Definition 5.6. Here $\Psi$ is a Poisson point process on $[0,\infty)\times(\Delta\setminus\{0\})$ with intensity $dt\,\langle x,x\rangle^{-1}\Xi(dx)$, encoding the times and sizes of GLIPs, and $c_{\mathrm{pair}}=1-\Xi(\Delta\setminus\{0\})$ is the limiting rate of binary mergers. Because a realized pedigree fixes which generations had large offspring numbers, those multiple-merger times are frozen in the quenched limit, whereas the annealed limit averages over them; hence the two limits differ exactly when $\Xi$ has atoms at nonzero offspring frequencies. The paper also shows that finite-dimensional distributions are the right notion of convergence: the prelimiting process spends time in states with two ancestral genes in one individual on the fast dispersal timescale, so it does not converge in the usual path-space topology, but its completely dispersed version does.

Load-bearing premise

The load-bearing premise is that before a generation with unusually large offspring numbers, ancestral gene copies are approximately uniformly scattered across the population's individuals—a step the paper explicitly labels a 'leap of faith' whose rigorous justification is left open, and on which the auxiliary $\varepsilon$-naive coalescent rests.

Editorial extensions

If this is right

  • If the theorem is right, the quenched and annealed limits agree only in the Kingman case $\Xi=\delta_0$; whenever the marginal coalescent has genuine multiple mergers, conditioning on the pedigree changes genealogies for every sample size.
  • The genealogy is fully described by a Poisson point process of large-family times plus a constant pair-merger rate; in the occasional-large-family example with $\gamma=1$ this is explicit: a $(\psi/4,\psi/4,\psi/4,\psi/4,0,\dots)$-merger at rate $4/(\psi^2+2)$ and Kingman mergers at rate $2/(\psi^2+2)$.
  • Multi-locus summaries inherit the pedigree effect: the paper's simulations show pedigree-specific bumps in the expected site-frequency spectrum and show via the law of total variance that the pedigree can explain a substantial fraction of variation in total tree length.
  • In the diploid Wright-Fisher model and finite-variance random-fitness models the quenched limit reduces to the Kingman coalescent, recovering the earlier Kingman result as a special case.
  • The coagulator-flow definition makes the limiting process meaningful even when potential jump times are dense, so the quenched limit does not require ordering infinitely many GLIPs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves open whether its 'leap of faith' in Section 4.3 can be upgraded to a theorem; if it can, the same limit should hold under weaker mixing assumptions, while a failure would require replacing formula (35) and Lemma 4.2.
  • A testable extension the paper only sketches is a quenched ancestral recombination graph for linked loci, with recombination rate $\rho$; the two-copy coupling developed here is the natural tool for that proof.
  • A statistical consequence the paper does not spell out is that shared GLIP times across unlinked loci make multi-locus site-frequency spectra informative about the times and sizes of past large families.
  • For the infinite-intensity Beta cases, the paper notes the simulation difficulty; a concrete check is whether truncating small paintbox events at a sample-size-dependent threshold preserves the quenched site-frequency spectrum.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the coalescent for a fixed diploid pedigree generated by the exchangeable bi-parental Cannings model of Birkner et al. (2018), conditioned on the pedigree. Under assumptions (6), (7), (12), and (13), it states that the rescaled pedigree-conditional n-coalescent converges to an inhomogeneous (Ψ, c_pair)-coalescent, where Ψ is a Poisson point process encoding the timing and scale of generations with large individual progeny and c_pair is the residual binary merger rate. The paper constructs the limiting process via a coagulator-based stochastic flow, proves the convergence by coupling two conditionally independent coalescents on the same pedigree, computes the generator limits in Section 6 and Appendix B, works through several examples including diploid Beta-coalescents, and illustrates pedigree effects on the site-frequency spectrum and total tree length.

Significance. If Theorem 3.8 is correct, this is a substantial contribution: it provides the first quenched coalescent limit for arbitrary sample size in a general diploid exchangeable population model, demonstrates a genuine quenched-annealed distinction when multiple mergers are present, and introduces a technically useful construction of inhomogeneous coalescents with dense potential jump times. The coagulator construction, the coupling strategy adapted from random walks in random environments, and the explicit examples are valuable and go well beyond previous sample-size-two or Kingman-only results. The paper is also commendably explicit about its main gap. However, because the proof of the central theorem depends on an unproved heuristic, the significance is currently conditional on closing that gap.

major comments (3)
  1. [Section 4.3, Eq. (35), Lemma 4.2] The main theorem is not established as written because the proof of Lemma 4.2 and the subsequent generator computations in Lemmas 6.10 and 6.11 rely on the 'leap of faith' stated in Section 4.3: before a generation with large individual progeny, ancestral gene positions are asserted to be approximately uniformly dispersed and independent of the pedigree. The authors explicitly write that they do not currently know how to make this rigorous. In the quenched setting, the positions X_j(g) are functions of the pedigree being conditioned on, so the passage from the left-hand side of (34) to the conditional law given V is exactly the point that needs proof; without it, the O(1/N) error in Lemma 4.2 is not justified. Since Lemma 4.2 feeds into the limits q_mix,epsilon and q_pure,epsilon that are used in Lemma 5.9 to control the squared L2 distance in (48), Theorem 3.8 currently rests on an unproved approximation rather than on the stated assumptions (6), (7), (12), and (13).
  2. [Appendix B, proof of Lemma 6.9] The asymptotic computation of q_pure in Lemma 6.9 is load-bearing, but its proof in Appendix B delegates several nontrivial facts to phrases such as 'a moment's thought' (after Eq. (72)) and 'it is not hard to see' (in the treatment of the case s > 0). These steps involve the limit of (N)_{d downarrow}/(c_N (N)_{b downarrow}) multiplied by products of falling factorials of bV_i, with K_l containing up to four contributions, and the combinatorial reorganization via the injections varrho. The recursion for the aggregate transition probabilities is stated without a fully written verification. Please provide complete details or a precise reference, because this lemma is one of the three generator limits on which Lemma 5.9 depends.
  3. [Section 5, N2_loc closure claim] The paragraph after Eq. (42) asserts that N2_loc is a closed subset of N under the stated weak convergence, arguing by mollifications of indicators of (0,u) x (Delta \ {0}). This is not immediate, because condition (40) involves an infinite sum of <x,x> and weak convergence of point processes does not in general preserve second-moment summability when atoms can accumulate near x = 0. Since the proof of Lemma 5.4 relies on the limiting configuration lying in N2_loc, please supply a proof of the closure claim or replace it with a topology that makes the domain closed.
minor comments (4)
  1. [Remark 3.10] The statement that cd(Π^{N,n}) converges in the Skorokhod topology is made without proof or reference; if this claim is retained, it should be proved or explicitly labeled as a conjecture.
  2. [Section 8] The simulation section would be more reproducible if the number of loci per pedigree, the number of pedigrees, and Monte Carlo standard errors were stated in the text rather than only in figure captions.
  3. [Throughout] There are numerous typographical and formatting artifacts, including 'c` adl` ag', 'L´ evy-Prokhorov', and several missing or scrambled diacritics in the references; a careful copyedit is needed.
  4. [Section 4.3] The phrase 'leap of faith' is appropriate, but the paper should make clear at the statement of Theorem 3.8 that the proof currently depends on this unproved heuristic; the current organization places this caveat only in the proof outline, which is easy to miss.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction; the explicit 'leap of faith' in Section 4.3 is a proof gap, not a circularity.

full rationale

The limiting parameters are not fitted: Ψ and cpair are built from the model's offspring measure through assumptions (13) and (18), and Theorem 3.8's conclusion is a quenched limit statement about the pedigree-conditional coalescent, not a restatement of those definitions. The proof uses a comparison to an auxiliary ε-naive coalescent (Definition 5.6, Lemma 5.9), and the generator computations in Lemmas 6.9–6.11 rely on the annealed results of Birkner et al. (2018); although that is a self-citation, it is prior independent mathematical work supplying the annealed limit rather than the quenched conclusion. The one caveat is Section 4.3: the authors write 'we presently do not know how to make it rigorous' about the uniform-dispersal approximation that enters Eq. (35), Lemma 4.2, and hence Lemmas 6.10–6.11. This is an explicitly acknowledged proof gap and a correctness risk for Theorem 3.8, but it is not a circular reduction: the approximation is not derived from the theorem, no fitted parameter is renamed as a prediction, and no equation equals its own input by construction. Thus no circular step is exhibited, and the score stays low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 3 invented entities

The theorem depends on the model assumptions (6), (7), (12), and (13), a no-selfing assumption, and a quoted separation-of-timescales lemma. The paper also explicitly relies, at least heuristically, on uniform dispersion of ancestral lines before GLIP events; although the coupling proof is designed to bypass this heuristic, the printed computation of the relevant generator rates, Lemma 6.9, is incomplete. No numerical parameters are fitted to data; model parameters in Section 7 are inputs illustrating the theory.

assumptions (6)
  • domain assumption Fixed population size and exchangeability of offspring matrix (V_ij) (assumptions (6), (7))
    Define the diploid Cannings model pedigree; invoked in Theorem 3.8 and throughout Sections 3 and 4.
  • domain assumption Rare large offspring: c_N → 0 (assumption (12))
    Ensures a non-degenerate time scale for coalescence; used to rescale time by 1/c_N.
  • domain assumption Vague convergence of rescaled offspring frequencies to Ξ' (assumption (13))
    Defines the input measure Ξ that parameterizes the limiting (Ψ,c)-coalescent; this is the model's large-offspring tail behavior, not derived.
  • ad hoc to paper Approximate uniform dispersion of ancestral genes before a GLIP (Section 4.3 'leap of faith')
    The authors state: 'We stress that this approximation is merely a heuristic and we presently do not know how to make it rigorous.' It motivates (35) and Lemma 4.2 which define the ε-naive coalescent.
  • standard math Separation of timescales result of Möhle (1998, Lemma 1 and Theorem 1)
    Quoted and applied in Section 6.2 to obtain generator convergence from aggregated transition matrices.
  • domain assumption No selfing (V_ii = 0)
    Explicit assumption of the model; ensures ancestral genes in the same individual disperse quickly.
invented entities (3)
  • Inhomogeneous (Ψ,c)-coalescent
    purpose: Limiting quenched coalescent process with pedigree-encoded multiple mergers at times given by Ψ and Kingman pair mergers at rate c.
    A new mathematical process, not a physical entity; its existence is the content of the paper rather than an independent hypothesis.
  • GLIP (generation with large individual progeny)
    purpose: Identifies generations whose large offspring families cause multiple-merger coalescent events.
    A descriptive label for an event type within the model; not a postulated physical mechanism.
  • Coagulator Coag_α
    purpose: Merges blocks of a partition according to a paintbox partition α, enabling a flow construction of the inhomogeneous coalescent with dense potential jump times.
    A mathematical operator introduced to handle dense jump times; no independent physical evidence applies.

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Pith. "Pith review of A conditional coalescent for diploid exchangeable population models given the pedigree." pith.science (2026). https://pith.science/paper/2T3GGQ6K

@misc{pith2026250515481,
  author       = {Pith},
  title        = {Pith review of: A conditional coalescent for diploid exchangeable population models given the pedigree},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2T3GGQ6K}},
  note         = {Machine review of arXiv:2505.15481}
}
abstract

We study coalescent processes conditional on the population pedigree under the exchangeable diploid bi-parental population model of \citet{BirknerEtAl2018}. While classical coalescent models average over all reproductive histories, thereby marginalizing the pedigree, our work analyzes the genealogical structure embedded within a fixed pedigree generated by the diploid Cannings model. In the large-population limit, we show that these conditional coalescent processes differ significantly from their marginal counterparts when the marginal coalescent process includes multiple mergers. We characterize the limiting process as an inhomogeneous $(\Psi,c)$-coalescent, where $\Psi$ encodes the timing and scale of multiple mergers caused by generations with large individual progeny (GLIPs), and $c$ is a constant rate governing binary mergers. Our results reveal fundamental distinctions between quenched (conditional) and annealed (classical) genealogical models, demonstrate how the fixed pedigree structure impacts multi-locus statistics such as the site-frequency spectrum, and have implications for interpreting patterns of genetic variation among unlinked loci in the genomes of sampled individuals. They significantly extend the results of \citet{DiamantidisEtAl2024}, which considered a sample of size two under a specific Wright-Fisher model with a highly reproductive couple, and those of \citet{TyukinThesis2015}, where Kingman coalescent was the limiting process. Our proofs adapt coupling techniques from the theory of random walks in random environments.

Figures

Figures reproduced from arXiv: 2505.15481 by the authors.

Figure 2
Figure 2. Two different genealogies on the same pedigree. The two figures above are two realizations of the Mendelian coin flips in the population of N = 6 individuals with the same pedigree. Consider a sample of n = 3 genes at generation g = 0. The sample Xi(0) 1⩽i⩽3 is described as follows: X1(0) = (0, 2) is the 0-gene of the second individual from the left, and X2(0) = (0, 5) and X3(0) = (1, 5) both belong to the fifth in… view at source ↗
Figure 3
Figure 3. Log base 10 of the expected SFS, specifically the expected proportion of polymorphic sites for each possible count of the mutant base in a sample of size n = 100, for three values of ψ ∈ {0.1, 0.5, 0.9} under the δ( ψ 4 , ψ 4 ,0,0,··· ) model with a Kingman component, in which λ is the rate of δ( ψ 4 , ψ 4 ,0,0,··· ) events relative to Kingman evens. In the top row (λ = 106 ) large reproduction events dominate Kingm… view at source ↗
Figure 4
Figure 4. Effects of pedigrees on the total length of the gene genealogy for n = 100 under the δ( 1 4 , 1 4 ,0,0,...) model with a negligible Kingman component (λ = 106 ). Left: Relative importance of the two sources of variation in Ttotal among loci from (60) expressed as fractions of the total variance, for values of ψ ranging from 0.025 to 1. The contributions sum to one, but both are displayed for illustration. Right: Dis… view at source ↗

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Works this paper leans on

77 extracted references · 64 canonical work pages

  1. [1]

    Maximum-likelihood estimation of demographic parameters using the frequency spectrum of unlinked single-nucleotide polymorphisms

    Alison M Adams and Richard R Hudson. Maximum-likelihood estimation of demographic parameters using the frequency spectrum of unlinked single-nucleotide polymorphisms. Genetics, 168 0 (3): 0 1699--1712, 2004. doi:10.1534/genetics.104.030171

    work page doi:10.1534/genetics.104.030171 2004

  2. [2]

    Exchangeability and related topics

    David J Aldous. Exchangeability and related topics. In P.L. Hennequin, editor, \' E cole d'\' E t\' e de Probabilit\' e s de Saint-Flour XXIII , volume 1117 of Lecture Notes in Mathematics. Springer-Verlag, 1985

    work page 1985

  3. [3]

    Gene genealogies within the organismal pedigrees of random-mating populations

    R Martin Ball, Joseph E Neigel, and John C Avise. Gene genealogies within the organismal pedigrees of random-mating populations. Evolution, 44 0 (2): 0 360--370, 1990. doi:10.1111/j.1558-5646.1990.tb05205.x

    work page arXiv 1990

  4. [4]

    Recent progress in coalescent theory

    Nathana \"e l Berestycki. Recent progress in coalescent theory. Ensaios Mathem \'a ticos , 16: 0 1--193, 2009. doi:10.21711/217504322009/em161

    work page doi:10.21711/217504322009/em161 2009

  5. [5]

    o hle, Steinr \

    Matthias Birkner, Jochen Blath, Martin M \"o hle, Steinr \"u cken, and Johanna Tams. A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks. ALEA Latin American Journal of Probability and Mathematical Statistics , 6: 0 35--61, 2009. URL https://alea.impa.br/articles/v6/06-02.pdf

    work page 2009

  6. [6]

    Statistical properties of the site-frequency spectrum associated with -coalescents

    Matthias Birkner, Jochen Blath, and Bjarki Eldon. Statistical properties of the site-frequency spectrum associated with -coalescents. Genetics, 195 0 (3): 0 1037--1053, 2013 a . doi:10.1534/genetics.113.156612

    work page doi:10.1534/genetics.113.156612 2013

  7. [7]

    An ancestral recombination graph for diploid populations with skewed offspring distribution

    Matthias Birkner, Jochen Blath, and Bjarki Eldon. An ancestral recombination graph for diploid populations with skewed offspring distribution. Genetics, 193 0 (1): 0 255--290, 2013 b . doi:10.1534/genetics.112.144329

    work page doi:10.1534/genetics.112.144329 2013

  8. [8]

    Directed random walk on the backbone of an oriented percolation cluster

    Matthias Birkner, Ji r \' i C ern\' y , Andrej Depperschmidt, and Nina Gantert. Directed random walk on the backbone of an oriented percolation cluster. Electronic Journal of Probability, 18: 0 1--35, 2013 c . doi:10.1214/EJP.v18-2302

    doi:10.1214/ejp.v18-2302 2013

Show all 77 references
  1. [9]

    Coalescent results for diploid exchangeable population models

    Matthias Birkner, Huili Liu, and Anja Sturm. Coalescent results for diploid exchangeable population models. Electronic Journal of Probability, 23: 0 1--44, 2018. doi:10.1214/18-EJP175

    2018 doi

  2. [10]

    On the fixation probability of an advantageous allele in a population with skewed offspring distribution

    Matthias Birkner, Florin Boenkost, Iulia Dahmer, and Cornelia Pokalyuk. On the fixation probability of an advantageous allele in a population with skewed offspring distribution. Electronic Journal of Probability, 29 0 (none): 0 1 -- 22, 2024. doi:10.1214/24-EJP1198

    2024 doi

  3. [11]

    On the static and dynamic points of view for certain random walks in random environment

    Erwin Bolthausen and Alain-Sol Sznitman. On the static and dynamic points of view for certain random walks in random environment. Methods and Applications of Analysis, 9 0 (3): 0 345--376, 2002. URL https://projecteuclid.org/journals/methods-and-applications-of-analysis/volume...

    2002

  4. [12]

    The hitchhiking effect on the site frequency spectrum of DNA polymorphisms

    John M Braverman, Richard R Hudson, Norman L Kaplan, Charles H Langley, and Wolfgang Stephan. The hitchhiking effect on the site frequency spectrum of DNA polymorphisms. Genetics, 140 0 (2): 0 783--796, 1995. doi:10.1093/genetics/140.2.783

    1995 doi

  5. [13]

    Gene frequency in small populations of mutant D rosophila

    Peter Buri. Gene frequency in small populations of mutant D rosophila. Evolution, 10 0 (4): 0 367--402, 1956. doi:10.1111/j.1558-5646.1956.tb02864.x

    1956

  6. [14]

    Genome-wide patterns of nucleotide polymorphism in domesticated rice

    Ana L Caicedo, Scott H Williamson, Ryan D Hernandez, Adam Boyko, Adi Fledel-Alon, Thomas L York, Nicholas R Polato, Kenneth M Olsen, Rasmus Nielsen, Susan R McCouch, Carlos D Bustamante, and Michael D Purugganan. Genome-wide patterns of nucleotide polymorphism in domesticated ...

    2007 doi

  7. [15]

    The latent roots of certain M arkov chains arising in genetics: a new approach

    C Cannings. The latent roots of certain M arkov chains arising in genetics: a new approach. I . H aploid models. Advances in Applied Probability, 6 0 (2): 0 260--290, 1974. doi:10.2307/1426293

    1974 doi

  8. [16]

    -Wright–Fisher processes with general selection and opposing environmental effects: F ixation and coexistence

    Fernando Cordero, Sebastian Hummel, and Gr \'e goire V \'e chambre. -Wright–Fisher processes with general selection and opposing environmental effects: F ixation and coexistence . The Annals of Applied Probability, 35 0 (1): 0 393 -- 457, 2025. doi:10.1214/24-AAP2117

    2025 doi

  9. [17]

    Pedigree in the biparental M oran model

    Camille Coron and Yves Le Jan. Pedigree in the biparental M oran model. Journal of Mathematical Biology, 84 0 (6): 0 51, 2022. doi:10.1007/s00285-022-01752-0

    2022 doi

  10. [18]

    The effect of consanguinity on coalescence times on the x chromosome

    Daniel J Cotter, Alissa L Severson, and Noah A Rosenberg. The effect of consanguinity on coalescence times on the x chromosome. Theoretical Population Biology, 140: 0 32--43, 2021. doi:10.1016/j.tpb.2021.03.004

    2021 doi

  11. [19]

    Generalized population models and the nature of genetic drift

    Ricky Der, Charles L Epstein, and Joshua B Plotkin. Generalized population models and the nature of genetic drift. Theoretical Population Biology, 80 0 (2): 0 80--99, 2011. doi:10.1016/j.tpb.2011.06.004

    2011 doi

  12. [20]

    Dynamics of neutral and selected alleles when the offspring distribution is skewed

    Ricky Der, Charles Epstein, and Joshua B Plotkin. Dynamics of neutral and selected alleles when the offspring distribution is skewed. Genetics, 191 0 (4): 0 1331--1344, 2012. doi:10.1534/genetics.112.140038

    2012 doi

  13. [21]

    Bursts of coalescence within population pedigrees whenever big families occur

    Dimitrios Diamantidis, Wai-Tong (Louis) Fan, Matthias Birkner, and John Wakeley. Bursts of coalescence within population pedigrees whenever big families occur . Genetics, 227 0 (1): 0 iyae030, 02 2024. ISSN 1943-2631. doi:10.1093/genetics/iyae030. URL https://doi.org/10.1093/g...

    2024 doi

  14. [22]

    Dual processes in population genetics

    Peter Donnelly. Dual processes in population genetics. In Petre Tautu, editor, Stochastic Spatial Processes, pages 94--105, Berlin, Heidelberg, 1986. Springer. doi:10.1007/BFb0076240

    1986 doi

  15. [23]

    Interpreting genetic variability: the effects of shared evolutionary history

    Peter Donnelly. Interpreting genetic variability: the effects of shared evolutionary history. In Derek Chadwick and Gail Cardew, editors, Variation in the Human Genome, Ciba Foundation Symposium 197, pages 25--40, Chichester, England, 1996. John Wiley & Sons, Ltd

    1996

  16. [24]

    A countable representation of the F leming- V iot measure-valued diffusion

    Peter Donnelly and Thomas G Kurtz. A countable representation of the F leming- V iot measure-valued diffusion. The Annals of Probability, 24 0 (2): 0 698 -- 742, 1996. doi:10.1214/aop/1039639359

    1996

  17. [25]

    Particle representations for measure-valued population models

    Peter Donnelly and Thomas G Kurtz. Particle representations for measure-valued population models. The Annals of Probability, 27 0 (1): 0 166--205, 1999. doi:10.1214/aop/1022677258

    1999

  18. [26]

    Sweepstakes reproduction facilitates rapid adaptation in highly fecund populations

    Bjarki Eldon and Wolfgang Stephan. Sweepstakes reproduction facilitates rapid adaptation in highly fecund populations. Molecular Ecology, 33 0 (10): 0 e16903, 2024. doi:10.1111/mec.16903

    2024 doi

  19. [27]

    Steven N. Evans. Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types. Ann. Inst. Henri Poincar \'e , Probab. Stat. , 33 0 (3): 0 339--358, 1997. ISSN 0246-0203. doi:10.1016/S0246-0203(97)80095-7. URL https://eudml.org/doc/77572

    1997 doi

  20. [28]

    A note on the sampling theory for infinite alleles and infinite sites models

    W J Ewens. A note on the sampling theory for infinite alleles and infinite sites models. Theoretical Population Biology, 6 0 (2): 0 143--148, 1974. doi:10.1016/0040-5809(74)90020-3

    1974 doi

  21. [29]

    Population genetics theory -- the past and the future

    Warren J Ewens. Population genetics theory -- the past and the future. In Sabin Lessard, editor, Mathematical and Statistical Developments of Evolutionary Theory, pages 177--227. Kluwer Academic Publishers, Amsterdam, 1990

    1990

  22. [30]

    Mathematical Population Genetics, Volume I: Theoretical Foundations

    Warren J Ewens. Mathematical Population Genetics, Volume I: Theoretical Foundations. Springer-Verlag, Berlin, 2004

    2004

  23. [31]

    Robust demographic inference from genomic and SNP data

    Laurent Excoffier, Isabelle Dupanloup, Emilia Huerta- S \'a nchez, Vitor C Sousa, and Matthieu Foll. Robust demographic inference from genomic and SNP data. PLOS Genetics , 9 0 (10): 0 1--17, 2013. doi:10.1371/journal.pgen.1003905

    2013 doi

  24. [32]

    The Genetical Theory of Natural Selection

    Ronald Aylmer Fisher. The Genetical Theory of Natural Selection. Clarendon, Oxford, 1930 a

    1930

  25. [33]

    The distribution of gene ratios for rare mutations

    Ronald Aylmer Fisher. The distribution of gene ratios for rare mutations. Proceedings of the Royal Society of Edinburgh, 50: 0 205--220, 1930 b . URL https://hdl.handle.net/2440/15106

    1930

  26. [34]

    Statistical properties of segregating sites

    Yun-Xin Fu. Statistical properties of segregating sites. Theoretical Population Biology, 48 0 (2): 0 172--197, 1995. doi:10.1006/tpbi.1995.1025

    1995

  27. [35]

    Statistical tests of neutrality of mutations

    Yun- X in X Fu and Wen- H siung Li. Statistical tests of neutrality of mutations. Genetics, 133 0 (3): 0 693--709, 1993. doi:10.1093/genetics/133.3.693

    1993 doi

  28. [36]

    Inference of super-exponential human population growth via efficient computation of the site frequency spectrum for generalized models

    Feng Gao and Alon Keinan. Inference of super-exponential human population growth via efficient computation of the site frequency spectrum for generalized models. Genetics, 202 0 (1): 0 235--245, 2016. doi:10.1534/genetics.115.180570

    2016 doi

  29. [37]

    The age of a mutation in a general coalescent tree

    Robert C Griffiths and Simon Tavar \' e . The age of a mutation in a general coalescent tree. Communications in Statistics. Stochastic Models, 14 0 (1-2): 0 273--295, 1998. doi:10.1080/15326349808807471

    1998 doi

  30. [38]

    Inferring the joint demographic history of multiple populations from multidimensional SNP frequency data

    Ryan N Gutenkunst, Ryan D Hernandez, Scott H Williamson, and Carlos D Bustamante. Inferring the joint demographic history of multiple populations from multidimensional SNP frequency data. PLOS Genetics , 5 0 (10): 0 1--11, 2009. doi:10.1371/journal.pgen.1000695

    2009 doi

  31. [39]

    First-passage time for many-particle diffusion in space-time random environments

    Jacob B Hass, Ivan Corwin, and Eric I Corwin. First-passage time for many-particle diffusion in space-time random environments. Physical Review E, 109: 0 054101, 2024. doi:10.1103/PhysRevE.109.054101

    2024 doi

  32. [40]

    Properties of a neutral allele model with intragenic recombination

    Richard R Hudson. Properties of a neutral allele model with intragenic recombination. Theoretical Population Biology, 23 0 (2): 0 183--201, 1983 a . doi:10.1016/0040-5809(83)90013-8

    1983 doi

  33. [41]

    Testing the constant-rate neutral allele model with protein sequence data

    Richard R Hudson. Testing the constant-rate neutral allele model with protein sequence data. Evolution, 37 0 (1): 0 203--217, 1983 b . doi:10.1111/j.1558-5646.1983.tb05528.x

    1983

  34. [42]

    Characterisation and convergence of random measures and point processes

    Olav Kallenberg. Characterisation and convergence of random measures and point processes. Probab. Theory Relat. Fields, 27 0 (1): 0 9--21, 1973

    1973

  35. [43]

    The number of heterozygous nucleotide sites maintained in a finite population due to the steady flux of mutations

    Motoo Kimura. The number of heterozygous nucleotide sites maintained in a finite population due to the steady flux of mutations. Genetics, 61 0 (4): 0 893--903, 1969. doi:10.1093/genetics/61.4.893

    1969 doi

  36. [44]

    The representation of partition structures

    J F C Kingman. The representation of partition structures. J. London Math. Soc., 18: 0 374--380, 1978

    1978

  37. [45]

    On the genealogy of large populations

    J F C Kingman. On the genealogy of large populations. Journal of Applied Probability, 19 0 (A): 0 27--43, 1982 a . doi:10.2307/3213548

    1982 doi

  38. [46]

    The coalescent

    J F C Kingman. The coalescent. Stochastic Processes and their Applications, 13 0 (3): 0 235--248, 1982 b . doi:10.1016/0304-4149(82)90011-4

    1982 doi

  39. [47]

    Exchangeability and the evolution of large populations

    J F C Kingman. Exchangeability and the evolution of large populations. In G Koch and F Spizzichino, editors, Exchangeability in Probability and Statistics, pages 97--112. North-Holland, Amsterdam, 1982 c

    1982

  40. [48]

    Common ancestors in a generalized M oran model, 2009

    Martin Linder. Common ancestors in a generalized M oran model, 2009. URL https://www.diva-portal.org/smash/get/diva2:310019/FULLTEXT01.pdf

    2009

  41. [49]

    A convergence theorem for M arkov chains arising in population genetics and the coalescent with selfing

    M M \" o hle. A convergence theorem for M arkov chains arising in population genetics and the coalescent with selfing. Advances in Applied Probability, 30 0 (2): 0 493--512, 1998. doi:10.1239/aap/1035228080

    1998

  42. [50]

    The concept of duality and applications to M arkov processes arising in neutral population genetics models

    Martin M \"o hle. The concept of duality and applications to M arkov processes arising in neutral population genetics models. Bernoulli, 5: 0 761--777, 1999

    1999

  43. [51]

    A classification of coalescent processes for haploid exchangeable population models

    Martin M \" o hle and Serik Sagitov. A classification of coalescent processes for haploid exchangeable population models. The Annals of Probability, 29 0 (4): 0 1547--1562, 2001. doi:10.1214/aop/1015345761

    2001

  44. [52]

    Coalescent patterns in diploid exchangeable population models

    Martin M \"o hle and Serik Sagitov. Coalescent patterns in diploid exchangeable population models. J. Math. Biol., 47 0 (4): 0 337--352, 2003. ISSN 0303-6812. doi:10.1007/s00285-003-0218-6

    2003 doi

  45. [53]

    Conditional gene genealogies given the population pedigree for a diploid moran model with selfing, 2024

    Maximillian Newman, John Wakeley, and Wai-Tong Louis Fan. Conditional gene genealogies given the population pedigree for a diploid moran model with selfing, 2024. URL https://arxiv.org/abs/2411.13048

    2024 arXiv

  46. [54]

    Estimation of population parameters and recombination rates from single nucleotide polymorphisms

    Rasmus Nielsen. Estimation of population parameters and recombination rates from single nucleotide polymorphisms. Genetics, 154 0 (2): 0 931--942, 2000. doi:10.1093/genetics/154.2.931

    doi

  47. [55]

    The coalescent process with selfing

    Magnus Nordborg and Peter Donnelly. The coalescent process with selfing. Genetics, 146 0 (3): 0 1185--1195, 1997. doi:10.1093/genetics/146.3.1185

    1997 doi

  48. [56]

    Exchangeable and partially exchangeable random partitions

    Jim Pitman. Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields, 102: 0 145--158, 1995

    1995

  49. [57]

    Coalescents with multiple collisions

    Jim Pitman. Coalescents with multiple collisions. The Annals of Probability, 27 0 (4): 0 1870--1902, 1999. doi:10.1214/aop/1022874819

    1902

  50. [58]

    An empirical approach to demographic inference with genomic data

    Peter L Ralph. An empirical approach to demographic inference with genomic data. Theoretical Population Biology, 127: 0 91--101, 2019. doi:10.1016/j.tpb.2019.03.005

    2019 doi

  51. [59]

    The general coalescent with asynchronous mergers of ancestral lines

    Serik Sagitov. The general coalescent with asynchronous mergers of ancestral lines. Journal of Applied Probability, 36 0 (4): 0 1116--1125, 1999. doi:10.1239/jap/1032374759

    1999

  52. [60]

    Convergence to the coalescent with simultaneous multiple mergers

    Serik Sagitov. Convergence to the coalescent with simultaneous multiple mergers. Journal of Applied Probability, 40 0 (4): 0 839--854, 2003. doi:10.1239/jap/1067436085

    2003

  53. [61]

    Experiments with the site frequency spectrum

    Raazesh Sainudiin, Kevin Thornton, Jennifer Harlow, James Booth, Michael Stillman, Ruriko Yoshida, Robert Griffiths, Gil McVean, and Peter Donnelly. Experiments with the site frequency spectrum. Bulletin of Mathematical Biology, 73 0 (4): 0 829--872, 2011. doi:10.1007/s11538-0...

    2011 doi

  54. [62]

    Population genetics of polymorphism and divergence

    Stanley A Sawyer and Daniel L Hartl. Population genetics of polymorphism and divergence. Genetics, 132 0 (4): 0 1161--1176, 1992. doi:10.1093/genetics/132.4.1161

    1992 doi

  55. [63]

    Coalescents with simultaneous multiple collisions

    Jason Schweinsberg. Coalescents with simultaneous multiple collisions. Electronic Journal of Probability, 5: 0 1--50, 2000. doi:10.1214/EJP.v5-68

    doi

  56. [64]

    Coalescent processes obtained from supercritical galton--watson processes

    Jason Schweinsberg. Coalescent processes obtained from supercritical galton--watson processes. Stochastic processes and their Applications, 106 0 (1): 0 107--139, 2003

    2003

  57. [65]

    A mutation rate model at the basepair resolution identifies the mutagenic effect of polymerase III transcription

    Vladimir Seplyarskiy, Evan M Koch, Daniel J Lee, Joshua S Lichtman, Harding H Luan, and Shamil R Sunyaev. A mutation rate model at the basepair resolution identifies the mutagenic effect of polymerase III transcription. Nature Genetics, 55: 0 2235--2242, 2023. doi:10.1038/s415...

    2023 doi

  58. [66]

    Variance and limiting distribution of coalescence times in a diploid model of a consanguineous population

    Alissa L Severson, Shai Carmi, and Noah A Rosenberg. Variance and limiting distribution of coalescence times in a diploid model of a consanguineous population. Theoretical Population Biology, 139: 0 50--65, 2021. doi:10.1016/j.tpb.2021.02.002

    2021 doi

  59. [67]

    On the meaning and existence of an effective population size

    P Sj \"o din, I Kaj, S Krone, M Lascoux, and M Nordborg. On the meaning and existence of an effective population size. Genetics, 169 0 (2): 0 1061--1070, 2005. doi:10.1534/genetics.104.026799

    2005 doi

  60. [68]

    Pairwise comparisons of mitochondrial DNA sequences in stable and exponentially growing populations

    Montgomery Slatkin and Richard R Hudson. Pairwise comparisons of mitochondrial DNA sequences in stable and exponentially growing populations. Genetics, 129 0 (2): 0 555--562, 1991. doi:10.1093/genetics/129.2.555

    1991 doi

  61. [69]

    Evolutionary relationship of DNA sequences in finite populations

    Fumio Tajima. Evolutionary relationship of DNA sequences in finite populations. Genetics, 105 0 (2): 0 437--460, 1983. doi:10.1093/genetics/105.2.437

    1983 doi

  62. [70]

    Statistical method for testing the neutral mutation hypothesis by DNA polymorphism

    Fumio Tajima. Statistical method for testing the neutral mutation hypothesis by DNA polymorphism. Genetics, 123: 0 585--595, 1989. doi:10.1093/genetics/123.3.585

    1989 doi

  63. [71]

    Lines-of-descent and genealogical processes, and their application in population genetic models

    S Tavar \'e . Lines-of-descent and genealogical processes, and their application in population genetic models. Theoretical Population Biology, 26 0 (2): 0 119--164, 1984. doi:10.1016/0040-5809(84)90027-3

    1984 doi

  64. [72]

    Quenched limits of coalescents in fixed pedigrees

    Andrey Tyukin. Quenched limits of coalescents in fixed pedigrees. Master's thesis, Johannes-Gutenberg-Universit\" a t Mainz, Germany, 2015. URL https://www.glk.uni-mainz.de/files/2018/08/andrey_tyukin_msc.pdf

    2015

  65. [73]

    Gene genealogies within a fixed pedigree, and the robustness of K ingman's coalescent

    John Wakeley, L\' e andra King, Bobbi S Low, and Sohini Ramachandran. Gene genealogies within a fixed pedigree, and the robustness of K ingman's coalescent. Genetics, 190 0 (4): 0 1433--1445, 2012. doi:10.1534/genetics.111.135574

    2012 doi

  66. [74]

    On the number of segregating sites in genetical models without recombination

    G A Watterson. On the number of segregating sites in genetical models without recombination. Theoretical Population Biology, 7 0 (2): 0 256--276, 1975. doi:10.1016/0040-5809(75)90020-9

    1975 doi

  67. [75]

    Sampling properties of genealogical pathways underlying population pedigrees

    Kurt Wollenberg and John C Avise. Sampling properties of genealogical pathways underlying population pedigrees. Evolution, 52 0 (4): 0 957--966, 1998. doi:10.1111/j.1558-5646.1998.tb01825.x

    1998

  68. [76]

    Evolution in M endelian populations

    Sewall Wright. Evolution in M endelian populations. Genetics, 16 0 (2): 0 97--159, 1931. URL https://www.genetics.org/content/16/2/97

    1931

  69. [77]

    Random walks in random environments

    Ofer Zeitouni. Random walks in random environments. Journal of Physics A: Mathematical and General, 39 0 (40): 0 R433, 2006. doi:10.1088/0305-4470/39/40/R01

    2006 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.