arXiv subjects

Florin Boenkost

Publications and source records attributed to Florin Boenkost.

9 recordsLinked to original sources

arXiv2026

Ornstein-Uhlenbeck fluctuations for the line counting process of the ancestral selection graph

For the Moran model with strong or moderate selection we prove that the fluctuations around the deterministic limit of the line counting process of the ancestral selection graph converge to an Ornstein-Uhlenbeck process. To this purpose we provide an extension of a functional limit theorem by Ethier and Kurtz 1986. This result and a small adaptation of our arguments can also be used to obtain the scaling limit for the fluctuations of certain logistic branching processes.

math.PR↗

arXiv2026

Logarithmic scaling of selective sweep curves: from tents to houses

One of the classical results of mathematical population genetics states that the frequency of a beneficial mutant's offspring, on its way to fixation in a large population, looks like a logistic curve. A logarithmic scaling (both in height and time) of these selective sweep curves leads (in the case of strong selection) to a tent-like shape in the large population limit: First the logarithmic frequency of the mutant increases linearly from 0 to 1, then that of the former resident decreases from 1 to 0. For moderate selection the logarithmic frequencies develop (in the large population limit) a jump at the beginning/the end of the sweep, which takes the shape of the tent into that of a house. Our main result (proved for the Moran model) assesses the regularity of this convergence in the large population limit: It is uniform in the house's roof (phases of linear growth and decline) and "Skorokhod $M_1$" in the house's walls (closely around the jumps). Apart from interest in its own right, we anticipate that this result and the proof techniques will be instrumental for extending the description of clonal interference by Poissonian interacting trajectories (as it was done in Hermann et al. (2024) for strong selection) also to moderate selection.

math.PR↗

arXiv2026

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

We consider a one-dimensional dyadic branching Brownian motion on $\mathbb{R}$ with positive drift $β\in (0,1)$, 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.

math.PR↗

arXiv2025

The genealogy of nearly critical branching processes in varying environment

Building on the spinal decomposition technique in Foutel-Rodier and Schertzer (2022) we prove a Yaglom limit law for the rescaled size of a nearly critical branching process in varying environment conditional on survival. In addition, our spinal approach allows us toprove convergence of the genealogical structure of the population at a fixed time horizon -- when the sequence of trees are envisioned as a sequence of metric spaces -- in the Gromov--Hausdorff--Prohorov (GHP) topology. We characterize the limiting metric space as a time-changed version of the Brownian coalescent point process Popovic (2004). Beyond our specific model, we derive several general results allowing one to go from spinal decompositions to convergence of random trees in the GHP topology. As a direct application, we show how this type of convergence naturally condenses the limit of several interesting genealogical quantities: the population size, the time to the most-recent common ancestor, the reduced tree, and the tree generated by $k$ uniformly sampled individuals. As in a recent article by the authors (Foutel-Rodier and Schertzer 2022), we hope that our specific example illustrates a general methodology that could be applied to more complex branching processes.

math.PR↗

arXiv2025

On the block size spectrum of a class of exchangeable dynamic random graphs

In this work we introduce the dynamic $Θ$-random graph and the associated $Θ$-coalescent with momentum. Dynamic $Θ$-random graphs are a subclass of exchangeable and consistent random graph processes, parametrised by a measure $Θ$ on $[0,1]\times (0,1]$, inspired by the classic $Λ$-coalescent from mathematical population genetics. The $Θ$-coalescent with momentum accounts for the small connected components of this graph; in contrast to the underlying random graph it is exchangeable but not consistent. Our main results specialise on the case where $Θ$ is the product of a beta measure and a Dirac mass at $1$. We prove a dynamic law of large numbers for the block size spectrum, which tracks the numbers of blocks containing $1,...,d$ elements. On top of that, we provide a functional limit theorem for the fluctuations. The limit process satisfies a stochastic differential equation of Ornstein-Uhlenbeck type.

math.PR↗

arXiv2024

Haldane's asymptotics for supercritical branching processes in an iid random environment

Branching processes in a random environment are natural generalisations of Galton-Watson processes. In this paper we analyse the asymptotic decay of the survival probability for a sequence of slightly supercritical branching processes in an iid random environment, where the offspring expectation converges from above to $1$. We prove that Haldane's asymptotics, known from classical Galton-Watson processes, turns up again in the random environment case, provided that one stays away from the critical/subcritical regime. A central building block is a connection to and a limit theorem for perpetuities with asymptotically vanishing interest rates.

math.PR↗

arXiv2024

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

Consider an advantageous allele that arises in a haploid population of size $N$ evolving in continuous time according to a skewed reproduction mechanism, which generates under neutrality genealogies lying in the domain of attraction of a Beta$(2-α, α)$-coalescent for $α\in (1,2)$. We prove in a setting of moderate selection that the fixation probability $π_N$ of the advantageous allele is asymptotically equal to $α^{1/(α-1)} s_N^{1/(α-1)} $ , where $s_N$ is the selection strength of the advantageous allele. Our proof uses duality with a suitable $Λ$-ancestral selection graph.

math.PR↗

arXiv2022

Haldane's formula in Cannings models: The case of moderately strong selection

For a class of Cannings models we prove Haldane's formula, $π(s_N) \sim \frac{2s_N}{ρ^2}$, for the fixation probability of a single beneficial mutant in the limit of large population size $N$ and in the regime of moderately strong selection, i.e. for $s_N \sim N^{-b}$ and $0< b<1/2$. Here, $s_N$ is the selective advantage of an individual carrying the beneficial type, and $ρ^2$ is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele's frequency process with slightly supercritical Galton-Watson processes in the early phase of fixation.

math.PR↗

arXiv2020

Haldane's formula in Cannings models: The case of moderately weak selection

We introduce a Cannings model with directional selection via a paintbox construction and establish a strong duality with the line counting process of a new \emph{Cannings ancestral selection graph} in discrete time. This duality also yields a formula for the fixation probability of the beneficial type. Haldane's formula states that for a single selectively advantageous individual in a population of haploid individuals of size $N$ the prob\-ability of fixation is asymptotically (as $N\to \infty$) equal to the selective advantage of haploids $s_N$ divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences $s_N$ obeying $N^{-1} \ll s_N \ll N^{-1/2} $, which is a regime of "moderately weak selection". It turns out that for $ s_N \ll N^{-2/3} $ the Cannings ancestral selection graph is so close to the ancestral selection graph of a Moran model that a suitable coupling argument allows to play the problem back asymptotically to the fixation probability in the Moran model, which can be computed explicitly.

math.PR↗