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The neutral Wright-Fisher model with $n$ individuals is a genealogical model often used in population genetics that can be described as follows: at all generations, there are exactly $n$ individuals, and each child chooses its parent uniformly at random. One can then follow genealogies backward in time.

In this simple model, one can take interest in the age of the most recent common ancestor $T_n$, defined as the first time when, looking for the ancestors of the ancestors of the ancestors... of all individuals of the first generation, only one ancestor remains.

My question is the following : is there a good estimate existing on the mean of $T_n$? I'm after not only an equivalent, but also a development up to $O(1)$ as $n \to \infty$.

For context, I'm aware of the Kingman coalescent, obtained when considering a finite subset of $k$ individuals in the current generation, and that writing $T_{n,k}$ for the age of the most recent common ancestor one has $$ \lim_{n \to \infty }T_{n,k}/n = \sum_{j=2}^k \frac{e_j}{j(j-1)} \quad \text{ in law.}$$ It hints that $\mathbb E(T_n)$ should grow at least as fast as $2n$.

I also tried to apply the result of Kimura and Osha (1969) on the fixation of a mutation in a Wright-Fisher process. I did not succeeded in applying it to this question as that result essentially considers the fixation of a mutation starting from $\epsilon n$ individuals instead of 1, and then letting $n \to \infty$ then $\epsilon \to 0$.

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