Let $f$ be a uniform random endomorphism on $\{1,2,...,n\}$.
We say $f$ is connected if its functional graph possesses exactly one connected component.
The unique cycle in this component has length $X_n$ and, according to Rényi (1959), $\lim_{n->\infty} X_n/\sqrt n$ follows the half-normal distribution.
Now suppose instead that the functional graph for $f$ possesses exactly two connected components.
Let $X_n \geq Y_n$ denote the lengths of the two associated cycles. What is the limiting distribution of $(X_n, Y_n)$? References would be welcome (very important to me)!