On the deepest cycle of a random mapping (Q6543047)

This paper provides asymptotic distributional results on the structure of a random mapping from a set of size \(n\) to itself. As any such mapping gives rise to a directed graph where every component has at most one cycle, a random mapping gives rise to a random directed graph that has this property. The main theorem of the paper provides the asymptotic distribution of the length of the cycle that is inside the largest component of the associated random digraph. It states that this random variable scaled by \(n^{1/2}\) converges in distribution as \(n\to \infty\) to a certain random variable that is given explicitly. Also, the asymptotic values of the expected value and the variance are given.