  An acyclic homomorphism of a digraph $C$ to a digraph $D$ is a function
$\rho\colon V(C)\to V(D)$ such that for every arc $uv$ of $C$, either
$\rho(u)=\rho(v)$, or $\rho(u)\rho(v)$ is an arc of $D$ and for every vertex
$v\in V(D)$, the subdigraph of $C$ induced by $\rho^{-1}(v)$ is acyclic. A
digraph $D$ is a core if the only acyclic homomorphisms of $D$ to itself are
automorphisms. In this paper, we prove that for certain choices of $p(n)$,
random digraphs $D\in D(n,p(n))$ are asymptotically almost surely cores. For
digraphs, this mirrors a result from [A. Bonato and P. Pra{\l}at, The good, the
bad, and the great: homomorphisms and cores of random graphs, Discrete Math.,
309 (2009), no. 18, 5535-5539; MR2567955] concerning random graphs and cores.
