A scaling limit for the length of the longest cycle in a sparse random digraph

We discuss the length L→c,n of the longest directed cycle in the sparse random digraph Dn,p,p=c/n , c constant. We show that for large c there exists a function f→(c) such that L→c,n/n→f→(c) a.s. The function f→(c)=1−∑k=1∞pk(c)e−kc where pk is a polynomial in c . We are only able to explicitly give the values p1,p2 , although we could in principle compute any pk .

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