The grapheme-valued Wright-Fisher diffusion with mutation

TL;DR

This paper introduces a new grapheme-valued diffusion process in population genetics, derived as a continuum limit of finite graph-based models with mutation, and characterizes its properties and stationary distribution.

Contribution

It provides the first example of a Markovian grapheme-valued diffusion model in population genetics with a rigorous convergence proof and links to the Poisson-Dirichlet distribution.

Findings

Grapheme-valued Markov chain converges to a diffusion as vertices grow large.

The stationary distribution of the diffusion relates to the Poisson-Dirichlet distribution.

The model extends the continuum limits of dense graphs in population genetics.

Abstract

In [Athreya, den Hollander, R\"ollin; 2021, arXiv:1908.06241] models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of dense graphs. In the present paper we exhibit an example of a simple neutral population genetics model for which this dynamics is a Markovian diffusion that can be characterised as the solution of a martingale problem. In particular, we consider a Markov chain in the space of finite graphs that resembles a Moran model with resampling and mutation. We encode the finite graphs as graphemes, which can be represented as a triple consisting of a vertex set, an adjacency matrix and a sampling measure. We equip the space of graphons with convergence of sample subgraph densities and show that the grapheme-valued Markov chain converges to a grapheme-valued diffusion as the number of vertices goes to infinity. We show that the grapheme-valued diffusion has a stationary distribution that is linked to the Poisson-Dirichlet distribution. In a companion paper [Greven, den Hollander, Klimovsky, Winter; 2023], we build up a general theory for obtaining grapheme-valued diffusions via genealogies of models in population genetics.