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arxiv: 2404.17527 · v3 · submitted 2024-04-26 · 🧮 math.PR

Power-law scaling of the effective population size in a branching particle system for moderate mutation-selection

Florin Boenkost , Julie Tourniaire This is my paper

Pith reviewed 2026-05-24 02:21 UTC · model grok-4.3

classification 🧮 math.PR
keywords branching Brownian motionKingman coalescentYaglom lawgenealogymutation-selection balanceeffective population sizecoalescent processespopulation genetics
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The pith

When particle cloud width scales as c log N, a branching model with deleterious mutations converges to Kingman's coalescent.

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

This paper studies a dyadic branching Brownian motion on the real line with positive drift, reflected at zero and killed at a position L tuned to maintain total population size around N. It serves as a model for a population under moderate selection against deleterious mutations. The authors establish that in the large N limit, for cloud widths of order c log N where c is between 0 and 1, the fluctuations in population size follow the Yaglom law over time scales that are polynomial in N. The genealogy in this regime is characterized by binary mergers occurring primarily near the reflecting boundary at zero, leading the system to belong to the universality class of Kingman's coalescent.

Core claim

In the large-N limit, when the typical width of the particle cloud is of order c log(N), with c ∈ (0,1), the demographic fluctuations follow a Yaglom law on a polynomial time scale. Moreover, the limiting genealogy of the system involves only binary mergers, concentrated near the reflecting boundary. Our system falls into the universality class of Kingman's coalescent.

What carries the argument

The one-dimensional dyadic branching Brownian motion with drift β ∈ (0,1), branching rate 1/2, reflected at 0 and killed at L chosen to keep population size ~N, which enforces cloud width c log N and produces binary mergers near the boundary.

If this is right

  • Demographic fluctuations follow a Yaglom law on polynomial time scales.
  • The limiting genealogy consists only of binary mergers concentrated near the reflecting boundary.
  • The model falls into the universality class of Kingman's coalescent.
  • Effective population size exhibits power-law scaling with N under this moderate mutation-selection regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The power-law scaling of effective population size follows directly from the Yaglom law and Kingman limit in the moderate selection window.
  • Varying the selection strength or drift parameter could shift the system between Kingman and other coalescent classes such as Bolthausen-Sznitman.
  • Direct simulation of the reflected branching Brownian motion at finite N could test the concentration of mergers near zero.

Load-bearing premise

The killing boundary L is chosen so that the total population size remains approximately constant, proportional to N.

What would settle it

If the genealogy exhibits non-binary mergers or mergers away from the reflecting boundary when the cloud width is c log N with c<1, the claim that the system belongs to the Kingman coalescent class would fail.

Figures

Figures reproduced from arXiv: 2404.17527 by Florin Boenkost, Julie Tourniaire.

Figure 1
Figure 1. Figure 1: k-spine tree with k = 3. Left panel: planar tree T generated from 2 i.i.d. uniform random variables (U1, U2). Right panel: branching 1-spines running along the branches of the tree T. Let T be the unique tree of depth t with k leaves such that the tree distance between the i-th and the j-th leaves is given Ui,j . This tree is ultrametric and planar in the sense that ∀i, j, l ∈ [k], Ui,j ≤ Ui,l ∨ Ul,j , (ul… view at source ↗
Figure 2
Figure 2. Figure 2: Intuitively, we expect our model to behave similarly to the following unstructured (i.e., non-spatial) model. Consider a Cannings model with population size N, where the offspring vector ν N = (ν N 1 , . . . , νN N ) in each generation is a random permutation of the vector v = ( N c , . . . , Nc | {z } N1−c coordinates , 0, . . . , 0). For simplicity, we assume that all powers are natural numbers. M¨ohle’s… view at source ↗
read the original abstract

We consider a one-dimensional dyadic branching Brownian motion on $\mathbb{R}$ with positive drift $\beta \in (0,1)$, branching rate $1/2$, reflected at $0$ and killed at a boundary $L > 0$. The killing boundary $L$ is chosen so that the total population size remains approximately constant, proportional to $N \in \mathbb{N}$. This branching process models a population accumulating deleterious mutations. In the large-$N$ limit, we prove that when the typical width of the particle cloud is of order $c \log(N)$, with $c \in (0,1)$, the demographic fluctuations follow a Yaglom law on a polynomial time scale. Moreover, the limiting genealogy of the system involves only binary mergers, concentrated near the reflecting boundary. Our model is a version of the branching Brownian motion with absorption introduced by Berestycki, Berestycki, and Schweinsberg to study the effect of beneficial mutations on genealogies. In sharp contrast with their model, whose genealogy is given by a Bolthausen--Sznitman coalescent, we show that our system falls into the universality class of Kingman's coalescent.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper considers a one-dimensional dyadic branching Brownian motion on R with positive drift β ∈ (0,1), branching rate 1/2, reflected at 0 and killed at a boundary L > 0 chosen so that the total population size remains approximately constant and proportional to N. This models a population accumulating deleterious mutations. In the large-N limit, when the typical width of the particle cloud is of order c log(N) with c ∈ (0,1), the authors prove that demographic fluctuations follow a Yaglom law on a polynomial time scale. The limiting genealogy involves only binary mergers concentrated near the reflecting boundary, placing the system in the universality class of Kingman's coalescent. This contrasts with the Bolthausen–Sznitman coalescent in the related model of Berestycki, Berestycki, and Schweinsberg for beneficial mutations.

Significance. If the stated convergence results hold, the work is significant for the study of genealogies in branching particle systems and population genetics models. It provides a precise scaling regime (cloud width c log N, c<1) that selects the Kingman coalescent over the Bolthausen–Sznitman coalescent, clarifying the effect of deleterious versus beneficial mutations on demographic fluctuations and merger structure. The tuned killing boundary to stabilize population size at scale N is a standard modeling device that enables the large-N analysis.

minor comments (2)
  1. The abstract refers to 'a polynomial time scale' without specifying the degree or the precise scaling; adding this detail would improve clarity for readers.
  2. Notation for the drift parameter β and the killing boundary L could be introduced with a brief reminder of their roles in the model definition section to aid readers unfamiliar with the Berestycki–Berestycki–Schweinsberg construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. The referee's summary correctly identifies the key results on the Yaglom demographic fluctuations and the emergence of the Kingman coalescent in the deleterious-mutation regime.

Circularity Check

0 steps flagged

No circularity; standard limit theorem from model definition

full rationale

The paper defines a reflected dyadic branching Brownian motion with drift, branching rate 1/2, and killing boundary L chosen to stabilize total population size at scale N. It then proves a Yaglom law for demographic fluctuations and Kingman coalescent genealogy when cloud width is c log N with c<1. These are direct consequences of the stochastic process construction and large-N analysis; no parameters are fitted to data, no predictions reduce to inputs by construction, and the cited contrast with Berestycki-Berestycki-Schweinsberg is to an external model by different authors. The derivation chain is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the model construction and standard properties of Brownian motion and branching processes; no free parameters are fitted to data, no new entities are postulated, and axioms are background facts from stochastic processes.

axioms (2)
  • standard math Standard properties of one-dimensional Brownian motion with drift, including reflection and absorption/killing, hold as in classical stochastic calculus.
    Invoked implicitly in the definition of the dyadic branching Brownian motion on R with reflection at 0 and killing at L.
  • domain assumption The large-N limit and scaling regime with cloud width c log N (c in (0,1)) can be analyzed via the chosen killing boundary that stabilizes population size at scale N.
    This is the key modeling premise stated in the abstract that enables the Yaglom law and Kingman convergence statements.

pith-pipeline@v0.9.0 · 5743 in / 1675 out tokens · 52023 ms · 2026-05-24T02:21:46.342780+00:00 · methodology

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