Edge disjoint Hamilton cycles in random digraphs of constant minimum degree

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\text{d}\ge k+1$, where $k\ge 1$. Denote such a random graph by $D_{n,m}^{(\delta \ge k+1)}$. Let $m=cn$ and $c\ge c_k$, where $c_k$ is a sufficiently large constant. We prove that w.h.p. $D_{n,m}^{(\delta \ge k+1)}$ contains $k$ edge disjoint Hamilton cycles.