Abstract
The genealogy at a single locus of a constant size $N$ population in equilibrium is given by the well-known Kingman's coalescent. When considering multiple loci under recombination, the ancestral recombination graph encodes the genealogies at all loci in one graph. For a continuous genome $\mathbb G$, we study the tree-valued process $(T^N_u)_{u\in\mathbb{G}}$ of genealogies along the genome in the limit $N\to\infty$. Encoding trees as metric measure spaces, we show convergence to a tree-valued process with cadlag paths. In addition, we study mixing properties of the resulting process for loci which are far apart.
Highlights
A large body of literature within the area of mathematical population genetics is dealing with models for population of constant size
The goal of the present paper is to study the sequence of genealogies along the genome, denoted, in the limit N → ∞
For a set of loci, called genome in the sequel, we aim to study the ancestry of individuals from a large population
Summary
A large body of literature within the area of mathematical population genetics is dealing with models for population of constant size While finite models such as the Wright-Fisher or the Moran model all have their specificities, the limit of large populations – given some moments are bounded – leads to a unified framework with diffusions and genealogical trees as their main tools; see e.g. We will use the notion of (ultra-) metric measure spaces, introduced in the probabilistic community by Greven et al (2009), in order to formalize genealogical trees, read off the sequence (TuN )u∈ from the ARG and let N → ∞. For I ⊆ Ê, the space DE(I) is the set of cadlag functions f : I → E
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
