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Kingman's coalescent is among the most fertile concepts in mathematical population genetics. However, it only approximates the exact coalescent process associated with the Wright-Fisher model, in which the ancestry of a sample does not have to be a binary tree. The distinction between the approximate and exact coalescent becomes important when population size is small and time has to be measured in discrete units (generations). In the present paper, we explore the exact coalescent, with mutations following the infinitely many sites model. The methods used involve random point processes and generating functionals. This allows obtaining joint distributions of segregating sites in arbitrary intervals or collections of intervals, and generally in arbitrary Borel subsets of two or more chromosomes. Using this framework it is possible to find the moments of the numbers of segregating sites on pairs of chromosomes, as well as the moments of the average of the number of pairwise differences, in the form that is more general than usually. In addition, we demonstrate limit properties of the first two moments under a range of demographic scenarios, including different patterns of population growth. This latter part complements results obtained earlier for Kingman's coalescent. Finally, we discuss various applications, including the analysis of fluctuation experiments, from which mutation rates of biological cells can be inferred.
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