Coalescent theory is the study of random processes where particles may join each other to
form clusters as time evolves. These notes provide an introduction to some aspects of the …

Our basic question is: Given a collection of DNA sequences, what underlying forces are
responsible for the observed patterns of variability? To approach this question we introduce …

This work reflects sixteen hours of lectures delivered by the author at the 2009 St Flour
summer school in probability. It provides a rapid introduction to a range of mathematical …

We study a family of coalescent processes that undergo “simultaneous multiple collisions,”
meaning that many clusters of particles can merge into a single cluster at one time, and …

Consider a population model in which there are N individuals in each generation. One can
obtain a coalescent tree by sampling n individuals from the current generation and following …

Contents. 1 Introduction 1.1 Genealogical processes 1.2 Organization of the notes 1.3
Acknowledgements 2 The Wright-Fisher model 2.1 Random drift 2.2 The genealogy of the …

We apply our general method of duality, introduced in Giardinà et al.(2007), to models of
population dynamics. The classical dualities between forward and ancestral processes can …

The Kingman coalescent, which has become the foundation for a wide range of theoretical
as well as empirical studies, was derived as an approximation of the Wright–Fisher (WF) …

We consider diploid bi-parental analogues of Cannings models: in a population of fixed size
$ N $ the next generation is composed of $ V_ {i, j} $ offspring from parents $ i $ and $ j …

For a class of Cannings models we prove Haldane's formula, π (s_N) ∼ 2s_N ρ^ 2 π (s N)∼
2 s N ρ 2, for the fixation probability of a single beneficial mutant in the limit of large …