Packing arborescences in random digraphs

We study the problem of packing arborescences in the random digraph $\mathcal D(n,p)$, where each possible arc is included uniformly at random with probability $p=p(n)$. Let $\lambda (\mathcal{D}(n,p))$ denote the largest integer $\lambda \ge 0$ such that, for all $0\le \mathrm{\ell }\le \lambda$, we have ${\sum }_{i=0}^{\mathrm{\ell }-1}(\mathrm{\ell }-i)\mathrm{\mid }\{v:d^{in}(v)=i\}\mathrm{\mid }\le \mathrm{\ell }$. We show that the maximum number of arc-disjoint arborescences in $\mathcal{D}(n,p)$ is $\lambda (\mathcal{D}(n,p))$ a.a.s. We also give tight estimates for $\lambda (\mathcal{D}(n,p))$ depending on the range of $p$.