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首页> 外文期刊>Current Science: A Fortnightly Journal of Research >Coalescence can be very recent in exploding populations
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Coalescence can be very recent in exploding populations

机译：爆炸人口中的合并可能是最近的

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摘要

Consider a binary family tree starting with one ancestor where each individual in any generation gives rise to two individuals in the next. In such a tree there will be 2n individuals in the nth generation. Now choose two of these 2n individuals of the nth generation at random and trace their lines of descent back in time till they meet, i.e. coalesce, and call that generation number χ n, the coalescence time. Then χ n is a random variable such that the probability χ n is less than k (for k ≥ 1) equals which is approximately (1 . b–k) which for n large. In particular, for any k ≥ 0, the probability that χ n equals k converges to b–k(1 . b–1). That is, for n large, coalescence of two randomly chosen individuals from the nth generation occurs in the remote past. This suggests that if the tree was generated by some random mechanism as long as it is growing rapidly, coalescence should occur in the remote past. It turns out that this need not hold if the growth is too rapid. In fact, the following result holds.
机译：考虑一个以一个祖先开始的二元家族树，其中任何一代中的每个个体都在下一个祖先中产生两个个体。在这样的树中，第n代将有2n个个体。现在，从第n代的这2n个人中随机选择两个，并追溯他们的下降路线，直到他们见面（即合并）为止，并将该代数χn称为合并时间。则χn是一个随机变量，因此概率χn小于k（对于k≥1）等于（大约为n的（1. b–k））。特别是，对于任何k≥0，χn等于k的概率收敛到b–k（1。b–1）。也就是说，对于n个大型对象，第n代中两个随机选择的个体的合并发生在遥远的过去。这表明，如果树是通过某种随机机制生成的，则只要树迅速生长，就应该在遥远的过去发生合并。事实证明，如果增长速度过快，则不必保持这种状态。实际上，以下结果成立。

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