On the number of Hamilton cycles in sparse random graphs

We prove that the number of Hamilton cycles in the random graph G(n, p) is n!pⁿ(1+ o(1))ⁿ asymptotically almost surely (a.a.s.), provided that p ≥ ln n+ln ln n+ω(1)/n. Furthermore, we prove the hitting time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree 2 creates (ln n/e) ⁿ(1 + o(1))ⁿ Hamilton cycles a.a.s.