Resilience for Loose Hamilton Cycles

We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum $d$-degree threshold for loose Hamiltonicity relative to the random $k$-uniform hypergraph $H_k(n,p)$ coincides with its dense analogue whenever $p\ge n^{-(k-1)\mathrm{/}2+o(1)}$. The value of $p$ is approximately tight for $d>(k+1)\mathrm{/}2$. This is particularly interesting because the dense threshold itself is not known beyond the cases when $d\ge k-2$.