The small subgraph conditioning method was introduced by Robinson and Wormald (1992, 1994) to prove that a random $r$-regular graph contains a Hamilton cycle with probability which tends to 1 as the number of vertices tends to infinity, so long as the degree $r$ is at least 3. The method involves a careful analysis of variance and has been applied to prove many other structural properties of random regular graphs. Much less work has been done in the hypergraph setting.
I will discuss recently-completed work on the analysis of the number of Hamilton cycles in random regular uniform hypergraphs. In particular, we find a constant degree threshold which is sufficient for the existence of a Hamilton cycle with probability which tends to 1, as conjectured by Dudek, Frieze, Rucinski and Sileikis in 2015.
This is joint work with Daniel Altman, Mikhail Isaev and Reshma Ramadurai.