Suppose you have $n,m$ simple molecules of two kinds $A,B$ that can be combined in three ways $AA,AB,BB$ (the molecules $AB$ and $BA$ are considered equal). At each step, choose uniformly two simple molecules that will be combined in a complex molecule (aka $AA$ will be produced with probability $p_{AA}(n,m)=\frac{n(n-1)}{(n+m)(n+m-1)}$, $AB$ with probability $2\frac{nm}{(n+m)(n+m-1)}$ and $BB$ with probability $\frac{m(m-1)}{(n+m)(n+m-1)}$).
Numerically - with Monte Carlo methods or calculating recursively the expected value for the complex molecules, for example $$E_{AA}(n,m) = p_{AA}(n,m)(1+E_{AA}(n-2,m)) \\ + p_{AB}(n,m)E_{AA}(n-1,m-1) + p_{BB}(n,m)E_{AA}(n,m-2)$$ it can be shown that $\frac{2E_{AA}(n,m)}{n+m} \sim (\frac{n}{n+m})^2$ for large $n,m$.
To restate the numerical observation, if $p$ is the proportion of $A$ and $q=1-p$ the proportion of B at the start of the process, at the end of the process the proportion of $AA$ is asymptotically $p^2$, of $AB$ is $2pq$ and of $BB$ is $q^2$.
This seems so simple and yet I can't find a lemma or exercise in the literature that states this observation...