Vertex-primitive digraphs with large fixity

The relative fixity of a digraph $\mathrm{\Gamma }$ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of $\mathrm{\Gamma }$ and the number of vertices of $\mathrm{\Gamma }$. We characterize the vertex-primitive digraphs whose relative fixity is at least $1\mathrm{/}3$, and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant.