First critical probability for a problem on random orientations in $G(n,p)$.

Abstract

We study the random graph $G(n,p)$ with a random orientation. For three fixed vertices $s,a,b$ in $G(n,p)$ we study the correlation of the events $\{a\to s\}$ (there exists a directed path from $a$ to $s$) and $\{s\to b\}$. We prove that asymptotically the correlation is negative for small $p$, $p<\frac{C_1}n$, where $C_1\approx0.3617$, positive for $\frac{C_1}n<p<\frac2n$ and up to $p=p_2(n)$. Computer aided computations suggest that $p_2(n)=\frac{C_2}n$, with $C_2\approx7.5$. We conjecture that the correlation  then stays negative for $p$ up to the previously known zero at $\frac12$; for larger $p$ it is positive.