  In the random hypergraph $H_{n,p;k}$ each possible $k$-tuple appears
independently with probability $p$. A loose Hamilton cycle is a cycle in which
every pair of adjacent edges intersects in a single vertex. We prove that if $p
n^{k-1}/\log n$ tends to infinity with $n$ then $$\lim_{\substack{n\to \infty
2(k-1) |n}}\Pr(H_{n,p;k}\ contains\ a\ loose\ Hamilton\ cycle)=1.$$ This is
asymptotically best possible.
