The threshold for loose Hamilton cycles in random hypergraph

We show that w.h.p.\ the random $r$-uniform hypergraph $H_{n,m}$ contains a loose Hamilton cycle, provided $r\ge 3$ and $m\ge \frac{(1+\epsilon )n\log n}{r}$, where $\epsilon$ is an arbitrary positive constant. This is asymptotically best possible, as if $m\le \frac{(1-\epsilon )n\log n}{r}$ then w.h.p.\ $H_{n,m}$ contains isolated vertices.