On the deepest cycle of a random mapping

Let $T_n$ be the set of all mappings $T:\{1,2,\dots ,n\}\to \{1,2,\dots ,n\}$. The corresponding graph of T is a union of disjoint connected unicyclic components. We assume that each $T\in T_n$ is chosen uniformly at random (i.e., with probability $n^{-n}$). The cycle of T contained within its largest component is called the deepest one. For any $T\in T_n$, let ${\nu }_n={\nu }_n\left(T\right)$ denote the length of this cycle. In this paper, we establish the convergence in distribution of ${\nu }_n/\sqrt{n}$ and find the limits of its expectation and variance as $n\to \infty$. For n large enough, we also show that nearly 55% of all cyclic vertices of a random mapping $T\in T_n$ lie in its deepest cycle and that a vertex from the longest cycle of T does not belong to its largest component with approximate probability 0.075.