  We establish a relation between two uniform models of random $k$-graphs (for
constant $k \ge 3$) on $n$ labeled vertices: $H(n,m)$, the random $k$-graph
with exactly $m$ edges, and $H(n,d)$, the random $d$-regular $k$-graph. By
extending to $k$-graphs the switching technique of McKay and Wormald, we show
that, for some range of $d = d(n)$ and a constant $c > 0$, if $m \sim cnd$,
then one can couple $H(n,m)$ and $H(n,d)$ so that the latter contains the
former with probability tending to one as $n \to \infty$. In view of known
results on the existence of a loose Hamilton cycle in $H(n,m)$, we conclude
that $H(n,d)$ contains a loose Hamilton cycle when $\log n = o(d)$ (or just $d
\ge C log n$, if $k = 3$) and $d = o(n^{1/2})$.
