The threshold for powers of tight Hamilton cycles in random hypergraphs

We investigate the occurrence of powers of tight Hamilton cycles in random hypergraphs. For every $r\ge 3$ and $k\ge 1$, we show that there exists a constant $C>0$ such that if $p=p(n)\ge C{n}^{-1\mathrm{/}\left(\genfrac{}{}{0px}{}{k+r-2}{r-1}\right)}$ then asymptotically almost surely the random hypergraph $H^{(r)}(n,p)$ contains the $k$th power of a tight Hamilton cycle. This improves on a result of Parczyk and Person, who proved the same result under the assumption $p=\omega \left(n^{-1\mathrm{/}\left(\genfrac{}{}{0px}{}{k+r-2}{r-1}\right)}\right)$ using a second moment argument.