Resilience for tight Hamiltonicity

We prove that random hypergraphs are asymptotically almost surely resiliently Hamiltonian. Specifically, for any $\gamma >0$ and $k\ge 3$, we show that asymptotically almost surely, every subgraph of the binomial random $k$-uniform hypergraph $G^{(k)}(n,n^{\gamma -1})$ in which all $(k-1)$-sets are contained in at least $(\frac{1}{2}+2\gamma )pn$ edges has a tight Hamilton cycle. This is a cyclic ordering of the $n$ vertices such that each consecutive $k$ vertices forms an edge.