Hamiltonian Berge cycles in random hypergraphs

In this note, we study the emergence of Hamiltonian Berge cycles in random $r$-uniform hypergraphs. For $r\ge 3$, we prove an optimal stopping-time result that if edges are sequently added to an initially empty $r$-graph, then as soon as the minimum degree is at least 2, the hypergraph almost surely has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erd\H{o}s–R'enyi random $r$-graph, and we also show that the $2$-out random $r$-graph almost surely has such a cycle. We obtain similar results for $\mathit{\text{weak}}\mathit{\text{Berge}}$ cycles as well, thus resolving a conjecture of Poole.