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

   
@article{EJP2725,
	author = {Sven Erick Alm and Svante Janson and Svante Linusson},
	title = {First critical probability for a problem on random orientations in $G(n,p)$.},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {19},
	year = {2014},
	keywords = {Random directed graph, correlation, directed paths},
	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.},
	pages = {no. 69, 1-14},
	issn = {1083-6489},
	doi = {10.1214/EJP.v19-2725},    
        url = {http://ejp.ejpecp.org/article/view/2725}}