The fixation line in the -coalescent

We define a Markov process in a forward population model with backward genealogy given by the -coalescent. This Markov process, called the fixation line, is related to the block counting process through its hitting times. Two applications are discussed. The probability that the n-coalescent is deeper than the (n - 1)-coalescent is studied. The distribution of the number of blocks in the last coalescence of the n-Beta(2.α,α)-coalescent is proved to converge as n → ∞, and the generating function of the limiting random variable is computed.