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

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 $(T_u^N{)}_{u\in G}$ of genealogies along the genome in the limit $N\to \mathrm{\infty }$. 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.