Abstract
Consider a population of fixed size that evolves over time. At each time, the genealogical structure of the population can be described by a coalescent tree whose branches are traced back to the most recent common ancestor of the population. As time goes forward, the genealogy of the population evolves, leading to what is known as an evolving coalescent. We will study the evolving coalescent for populations whose genealogy can be described by the Bolthausen Sznitman coalescent. We obtain the limiting behavior of the evolution of the time back to the most recent common ancestor and the total length of the branches in the tree. By similar methods, we also obtain a new result concerning the number of blocks in the Bolthausen-Sznitman coalescent.
Highlights
Consider a haploid population of fixed size n that evolves over time
The genealogy of the population at time t can be represented by a coalescent process (Π(s), s ≥ 0) taking its values in the set of partitions of {1, . . . , n}, which is defined so that integers i and j are in the same block of Π(s) if and only if the ith and jth individuals in the population at time t have the same ancestor at time t − s
The goal of the present paper is to determine the dynamics of the time back to the most recent common ancestor (MRCA) and the total branch length for populations whose genealogy is given by the Bolthausen-Sznitman coalescent
Summary
Consider a haploid population of fixed size n that evolves over time. The genealogy of the population at time t can be represented by a coalescent process (Π(s), s ≥ 0) taking its values in the set of partitions of {1, . . . , n}, which is defined so that integers i and j are in the same block of Π(s) if and only if the ith and jth individuals in the population at time t have the same ancestor at time t − s. The associated evolving coalescent was studied by Pfaffelhuber and Wakolbinger [23] They showed that the jumps of the process (A(t), t ≥ 0) that follows the time back to the MRCA occur at times of a homogeneous Poisson process, but that the process (A(t), t ≥ 0) is not Markov. The goal of the present paper is to determine the dynamics of the time back to the MRCA and the total branch length for populations whose genealogy is given by the Bolthausen-Sznitman coalescent. The questions about the evolution of the time back to the MRCA and the total branch length could be considered for other coalescents with multiple mergers besides the Bolthausen-Sznitman coalescent.
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