Hamilton cycles in random digraphs with minimum degree at least one

We study the existence of a directed Hamilton cycle in random digraphs with $m$ edges where we condition on minimum in- and out-degree at least one. Denote such a random graph by $D_{n,m}^{(\delta \ge 1)}$. We prove that if $m=\frac{n}{2}(\log n+2\log \log n+c_n)$ then $$\underset{n\to \mathrm{\infty }}{\lim }\mathrm{Pr}(D_{n,m}^{(\delta \ge 1)}\text{ is Hamiltonian})=\{\begin{array}{ll}0& c_n\to -\mathrm{\infty }\mathrm{.}\\ e^{-e^{-c}\mathrm{/}4}& c_n\to c\mathrm{.}\\ 1& c_n\to \mathrm{\infty }\mathrm{.}\end{array}$$