### On classifying genealogies for general (diploid) exchangeable population models

Probability Seminar

9th February 2018, 3:30 pm – 4:30 pm

Main Maths Building, SM4

The genetic variation in a sample of individuals/genes depends on their relatedness

which is described by their genealogy. In this talk we consider classifying the genealogies

of exchangeable population models with fixed size N asymptotically as N tends to infinity.

In the first part of the talk we review classical results regarding the haploid Cannings model. Here, each individual

is represented by one of its genes and thus each offspring (gene) has a unique parent. In each generation,

the offspring vector to the N parents is exchangeable. With an appropriate rescaling the corresponding

coalescence processes describing the genealogy converge to a limit process. Möhle and Sagitov (2001) classified

all possible limit processes and showed that depending on the tail behavior of the offspring numbers the limit

process is Kingman's coalescent with coalescence of pairs or is given by coalescents with (simultaneous)

multiple mergers in which (several) larger groups may find a common ancestor at the same time.

In the second part of the talk we extend this result to diploid bi-parental analogues of the Cannings model.

Here, the next generation is composed of offspring of parent pairs, which form an exchangeable (symmetric) array.

Also, every individual carries two gene copies, each of which is inherited from one of its parents. Our result

classifies the limiting coalescent processes describing the gene genealogies. Using this general result we determine

the limiting coalescent in a number of examples of which some have been studied previously (in special cases)

and some are new. We also point out connections to the theory of random graphs.

This talk is based on joint work with Matthias Birkner (Universit\"at Mainz) and Huili Liu (Hebei Normal University).

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