Hannah Dopmeyer | Searcharxiv

arXiv2026

Genealogical structures under interactive neutral reproduction: factorial moment duality via a Frankenstein process

We establish a genealogical framework for an existing analytical moment duality between a Wright--Fisher type SDE and a counting process with interaction. To achieve this, we construct a finite-population Moran model featuring interactive neutral reproduction as a novel mechanism. In the corresponding events, an individual, regardless of its own type, can only reproduce if a randomly encountered partner is of the ``fit'' type. This Moran model has a relatively simple counting process as its factorial moment dual, whose genealogical meaning appears to be cryptic: after all, the line-counting process of the natural genealogical process of the model, namely the ancestral influence graph (AIG), exhibits a complex hierarchical structure not reflected in the factorial moment dual. Since moment duality is a property in expectation, we are allowed to systematically remove information from the AIG and merge different realizations of the ancestry. We call the result the \emph{Frankenstein process}. Based on this, we establish the factorial moment duality from a genealogical perspective. The moment duality in the diffusion limit follows in a natural way.

math.PR↗

arXiv2025

A two-size Wright--Fisher model: asymptotic analysis via uniform renewal theory

We consider a population with two types of individuals, distinguished by the resources required for reproduction: type-$0$ (small) individuals need a fractional resource unit of size $\vartheta \in (0,1)$, while type-$1$ (large) individuals require $1$ unit. The total available resource per generation is $R$. To form a new generation, individuals are sampled one by one, and if enough resources remain, they reproduce, adding their offspring to the next generation. The probability of sampling an individual whose offspring is small is $\rho_{R}(x)$, where $x$ is the proportion of small individuals in the current generation. We call this discrete-time stochastic model a two-size Wright--Fisher model, where the function $\rho_{R}$ can represent mutation and/or frequency-dependent selection. We show that on the evolutionary time scale, i.e. accelerating time by a factor $R$, the frequency process of type-$0$ individuals converges to the solution of a Wright--Fisher-type SDE. The drift term of that SDE accounts for the bias introduced by the function $\rho_{R}$ and the consumption strategy, the latter also inducing an additional multiplicative factor in the diffusion term. To prove this, the dynamics within each generation are viewed as a renewal process, with the population size corresponding to the first passage time $\tau(R)$ above level $R$. The proof relies on methods from renewal theory, in particular a uniform version of Blackwell's renewal theorem for binary, non-arithmetic random variables, established via $\varepsilon$-coupling.

math.PR↗