A mixing tree-valued process arising under neutral evolution with recombination

Andrej Depperschmidt (1) , Étienne Pardoux (2) , Peter Pfaffelhuber (1)

1 Abteilung für Mathematische Stochastik
2 I2M - Institut de Mathématiques de Marseille

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 G, we study the tree-valued process (T N u)u∈G of genealogies along the genome in the limit N → ∞. Encoding trees as metric measure spaces, we show convergence to a tree-valued process with càdlàg paths. In addition, we study mixing properties of the resulting process for loci which are far apart.

Mots clés: Ancestral recombination graph; Kingman coalescent; tree-valued process; Gromov-Hausdorff metric

Domaines: Mathématiques [math]; Mathématiques générales [math.GM]

Andrej Depperschmidt, Étienne Pardoux, Peter Pfaffelhuber. A mixing tree-valued process arising under neutral evolution with recombination. Electronic Journal of Probability, 2015, 20, pp.1-22. ⟨10.1214/EJP.v20-4286⟩. ⟨hal-01237957⟩