  For positive integers $r > \ell$, an $r$-uniform hypergraph is called an
$\ell$-cycle if there exists a cyclic ordering of its vertices such that each
of its edges consists of $r$ consecutive vertices, and such that every pair of
consecutive edges (in the natural ordering of the edges) intersect in precisely
$\ell$ vertices. Such cycles are said to be linear when $\ell = 1$, and
nonlinear when $\ell > 1$. We determine the sharp threshold for nonlinear
Hamiltonian cycles and show that for all $r > \ell > 1$, the threshold $p^*_{r,
\ell} (n)$ for the appearance of a Hamiltonian $\ell$-cycle in the random
$r$-uniform hypergraph on $n$ vertices is sharp and is $p^*_{r, \ell} (n) =
\lambda(r,\ell) (\frac{\mathrm{e}}{n})^{r - \ell}$ for an explicitly specified
function $\lambda$. This resolves several questions raised by Dudek and Frieze
in 2011.
