13:15
I will take as my starting point a problem which is classical in
population genetics: we wish to understand the distribution of numbers
of individuals in a population who carry different alleles of a
certain gene. We imagine a sample of size n from a population in
which individuals are subject to neutral mutation at a certain
constant rate. Every mutation gives rise to a completely new type.
The genealogy of the sample is modelled by a coalescent process and we
imagine the mutations as a Poisson process of marks along the
coalescent tree. The allelic partition is obtained by tracing back to
the most recent mutation for each individual and grouping together
individuals whose most recent mutations are the same. The number of
blocks of each of the different possible sizes in this partition is
called the allele frequency spectrum. Recently, there has been much
interest in this problem when the underlying coalescent process is a
so-called Lambda-coalescent (even when this is not a biologically
``reasonable'' model) because the allelic partition is a nice example
of an exchangeable random partition. In this talk, I will describe
the asymptotics (as n tends to infinity) of the allele frequency
spectrum when the coalescent process is a particular Lambda-coalescent
which was introduced by Bolthausen and Sznitman. It turns out that
the frequency spectrum scales in a rather unusual way, and that we
need somewhat unusual tools in order to tackle it.
This is joint work with Anne-Laure Basdevant (Toulouse III).