Wright-Fisher model and Kingman Coalescence

In the last course of ecology and model of probability at Polytechnique, we talk about Wright-Fisher model and Kingman coalescence, two models used for the simulation of the genes of human beings. The former is easy to understand and the latter, relates the theory of combinatoire, provides some very interesting formulas.

Wright-Fisher model

Suppose that in the population there exists two types of genes A and a, then we denote $X_n$ the number of A in the population, whose size is always $N$. Then the evolution is a Markov chain and the transition matrix is a binomial type
$$\mathbb{P}(X^N_{n+1} =  k| X^N_{n} = i) = C_N^k \left(\frac{i}{N}\right)^k \left(1 - \frac{i}{N}\right)^{N-i}$$
I
It is easy to check that $X^N_n$ is a martingale and its $L^2$ norm is bounded and the Markov chain is positive recurrent, so
$$X^N_n \xrightarrow[a.s]{n \rightarrow \infty} X^N_{\infty} \in \{0, N\}$$
Using the theorem of stopping time we get that $\mathbb{P}(X^N_{\infty} = N) = \frac{i}{N}$ where $i$ is the initial state.

Some other version can also be developped like the model with mutation and selection. On another hand,  if we change the scale of time like $Z_t = \frac{1}{N}X^N_{[Nt]}$, the convergence of trajectoire implies that
$$Z_t = Z_0 + \int_0^t \sqrt{Z_s (1 - Z_s)} dB_s$$
which relates a discrete model with a continuous random process.

Kingman coalescence model

The state is defined on the partition of $[1,N]$ and the initial state is $X_0 = \{1\}\{2\}\cdots\{N\}$. We note $T_i$ the i-th jump time which follows the law $\mathcal{Exp}(\frac{(N-i)(N-i-1)}{2})$ and choose uniformly two block to make one block. Some obvious properties are observed.

1. Each time, the number of block minus one.

2. The expectation of fusion time is $\sum_{i=1}^{N-1} \frac{2}{(N-i)(N-i-1)} = 2(1 - \frac{1}{N-1})$ and it converges.

3. We define the genetic relation $\pi' \rightarrow \pi$ if the later is the generated by fusing two blocks of the former. The transition matrix is 

$$\mathbb{P}(\pi', \pi) = \frac{2}{\sharp \pi' \cdot (\sharp \pi' -1)} \mathbb{I}_{\pi' \rightarrow \pi}$$

However, the most beautiful formule is 

$$\mathbb{P}_{T_{N-k}}(\pi) = \frac{k !}{N !} \frac{(N-k)!(k - 1)!}{(N-1)!} \prod_{i = 1}^l \sharp B_i$$

for $\pi = \bigsqcup_{i=1}^k \{B_i\}$

The proof is just  a recurrence but the structure of formula is really beautiful, isn't?