The fixation line in the ${\Lambda}$-coalescent

We define a Markov process in a forward population model with backward genealogy given by the $\Lambda$-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 $n$-$\operatorname {Beta}(2-\alpha,\alpha)$-coalescent is proved to converge as $n\rightarrow\infty$, and the generating function of the limiting random variable is computed.