Locality of Random Digraphs on Expanders

We study random digraphs on sequences of expanders with bounded average degree and weak local limit. The threshold for the existence of a giant strongly connected component, as well as the asymptotic fraction of nodes with giant fan-in or giant fan-out are local, in the sense that they are the same for two sequences with the same weak local limit. The digraph has a bow-tie structure, with all but a vanishing fraction of nodes lying either in the unique strongly connected giant and its fan-in and fan-out, or in sets with small fan-in and small fan-out. All local quantities are expressed in terms of percolation on the limiting rooted graph, without any structural assumptions on the limit, allowing, in particular, for non tree-like limits. In the course of proving these results, we prove that for unoriented percolation, there is a unique giant above criticality, whose size and critical threshold are again local. An application of our methods shows that the critical threshold for bond percolation and random digraphs on preferential attachment graphs is $p_c=0$, with an infinite order phase transition at $p_c$.