Aspects of exchangeable coalescent processes - Pitters, HH (2015)
In mathematical population genetics a multiple merger n-coalescent process, or Λ n-coalescent process, {Πn(t) t ≥ 0} models the genealogical tree of a sample of size n (e.g. of DNA sequences) drawn from a large population of haploid individuals. We study various properties of Λ coalescents. Novel in our approach is that we introduce the partition lattice as well as cumulants into the study of functionals of coalescent processes. We illustrate the success of this approach on several examples. Cumulants allow us to reveal the relation between the tree height, Tn, respectively the total branch length, Ln, of the genealogical tree of Kingman’s n-coalescent, arguably the most celebrated coalescent process, and the Riemann zeta function. Drawing on results from lattice theory, we give a spectral decomposition for the generator of both the Kingman and the Bolthausen-Sznitman n-coalescent, the latter of which emerges as a genealogy in models of populations undergoing selection. Taking mutations into account, let Mj count the number of mutations that are shared by j individuals in the sample. The random vector (M1,...,Mn-1), known as the site frequency spectrum, can be measured from genetical data and is therefore an important statistic from the point of view of applications. Fu worked out the expected value, the variance and the covariance of the marginals of the site frequency spectrum. Using the partition lattice we derive a formula for the cumulants of arbitrary order of the marginals of the site frequency spectrum. Following another line of research, we provide a law of large numbers for a family of Λ coalescents. To be more specific, we show that the process {#Πn(t), t ≥ 0} recording the number #Πn(t) of individuals in the coalescent at time t, coverges, after a suitable rescaling, towards a deterministic limit as the sample size n grows without bound. In the statistical physics literature this limit is known as a hydrodynamic limit. Up to date the hydrodynamic limit was known for Kingman’s coalescent, but not for other Λ coalescents. We work out the hydrodynamic limit for beta coalescents that come down from infinity, which is an important subclass of the Λ coalescents.