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Recent progress in coalescent theory. (English) Zbl 1204.60002

Ensaios Matemáticos 16. Rio de Janeiro: Sociedade Brasileira de Matemática (ISBN 978-85-85818-40-1/pbk). 193 p. (2009).
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 mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks. The stated purpose of this book is to give a quick introduction to the underlying mathematical ideas, and to the biological motivations. The book is self-contained assuming knowledge of Poisson processes and Brownian motion as prerequisites. The focus is on ideas and accessibility, possibly at the price of mathematical rigor, and the exposition of this demanding area of probability theory is lucid and a pleasure to read.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J25 Continuous-time Markov processes on general state spaces
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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