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

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

2009-06-03

arxiv.org/abs/0906.0720v2

Mathematics

Probability

Author added, main proof greatly simplified and extended to cover also G(n,m). Discussion on quenched version added

Scientific paper

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 combined probability space, of the events a -> s and s -> b. For G(n,p), we prove that there is a p_c=1/2 such that for a fixed pp_c 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 $p=m/\binom{n}{2}$, there is a critical p_c 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 k directed edges in G(n,m), is thought to be of independent interest. We present exact recursions to compute P(a -> s)$ and P(a -> s, s -> b)$. We also briefly discuss the corresponding question in the quenched version of the problem.

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