Hitting-time threshold for loose Hamilton cycles

Establish a hitting-time version of the threshold theorem for loose Hamilton cycles in random uniform hypergraphs, determining whether the first moment at which the minimum-degree condition of rho > 2 is met coincides asymptotically with the first occurrence of a loose Hamilton cycle.

Background

The paper establishes the asymptotically sharp edge threshold for the appearance of loose Hamilton cycles in random r-uniform hypergraphs, using a construction involving random subhypergraphs and perfect matchings. Previous work of Kahn established a hitting-time result for perfect matchings, suggesting an analogous question for loose Hamilton cycles.

The authors explain that their current method requires a minimum degree exceeding the constant rho(r) from the random-regular-hypergraph loose Hamiltonicity theorem. They state that it is not evident how to adapt or tighten the approach to obtain a hitting-time result, leaving the relationship between the minimum-degree hitting time and the loose-Hamilton-cycle hitting time unresolved.