Consider the set of all digraphs on @math with @math edges, whose minimum in-degree and minimum out-degree are at least @math and @math respectively. For @math and @math , @math , we show that, among those digraphs, the fraction of @math -strongly connected digraphs is @math . Earlier with Dan Poole we identified a sharp edge-density threshold @math for birth of a giant @math -core in the random digraph @math . Combining the claims, for @math with probability @math the giant @math -core exists and is @math -strongly connected.
The upshot of this discussion is that, conditioned on the terminal vertex set and the terminal number of edges, the terminal sequence @math is distributed uniformly. So Theorem in combination with Theorem from @cite_15 yield
@cite_15
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The @math -core of a digraph is the largest sub-digraph with minimum in-degree and minimum out-degree at least @math and @math respectively. For @math , we establish existence of the threshold edge-density @math , such that the random digraph @math , on the vertex set @math with @math edges, asymptotically almost surely has a giant @math -core if @math , and has no @math -core if @math . Specifically, denoting @math by @math , we prove that @math .