  Let $\mathcal{T}_n$ be the set of all mappings
$T:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$. The corresponding graph of $T$ is a
union of disjoint connected unicyclic components. We assume that each
$T\in\mathcal{T}_n$ is chosen uniformly at random (i.e., with probability
$n^{-n}$). The cycle of $T$ contained within its largest component is callled
the deepest one. For any $T\in\mathcal{T}_n$, let $\nu_n=\nu_n(T)$ 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 $T\in\mathcal{T}_n$ lie in the
deepest cycle and that a vertex from the longest cycle of $T$ does not belong
to its largest component with approximate probability $0.075$.
