Correlations for paths in random orientations of G(n,p) and G(n,m)

We study random graphs, both G( n,p) and G( n,m), with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combine probability space, of the events $\{a\to s\}$ and $\{s\to b\}$ . For G(n,p) , we prove that there is a $pc = 1/2$ such that for a fixed $p < pc$ the correlation is negative for large enough n and for $p > pc$ the correlation is positive for large enough n . We conjecture that for a fixed $n \ge 27$ the correlation changes sign three times for three critical values of p . For G(n,m) it is similarly proved that, with $p=m/({{n}\atop {2}})$ , there is a critical pc that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any $\ell$ directed edges in G(n,m) , is thought to be of independent interest. We present exact recursions to compute \input amssym $\Bbb{P}(a\to s)$ and \input amssym $\Bbb{P}(a\to s, s\to b)$ . We also briefly discuss the corresponding question in the quenched version of the problem. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011