For p ≥ log^15 n / n, the random digraph D_{n,p} almost surely contains exactly δ±(D_{n,p}) edge-disjoint Hamilton cycles.
Bollobás,The evolution of sparse graphs, Graph theory and combinatorics (Cambridge, 1983) (1984), 35–57
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Minimum degree edge-disjoint Hamilton cycles in random directed graphs
For p ≥ log^15 n / n, the random digraph D_{n,p} almost surely contains exactly δ±(D_{n,p}) edge-disjoint Hamilton cycles.