The almost-spanning cycle conjecture for sparse random hypercube subgraphs

Let $\mathcal{Q}^n$ be the $n$-dimensional hypercube, and let $\mathcal{Q}^n_p$ be the random subgraph obtained by retaining each edge independently with probability $p=p(n)$. The almost-spanning cycle conjecture. If $pn\to\infty$, then asymptotically almost surely $\mathcal{Q}^n_p$ contains a cycle of length $(1-o(1))2^n$. The preceding theorem proves the corresponding assertion for fixed $p>0$ and any fixed positive loss in the proportion of vertices; the conjecture asks for an almost-spanning cycle throughout the wider sparse range.