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2014 First critical probability for a problem on random orientations in $G(n,p)$.
Sven Erick Alm, Svante Janson, Svante Linusson
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Electron. J. Probab. 19(none): 1-14 (2014). DOI: 10.1214/EJP.v19-2725

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.

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Sven Erick Alm. Svante Janson. Svante Linusson. "First critical probability for a problem on random orientations in $G(n,p)$.." Electron. J. Probab. 19 1 - 14, 2014. https://doi.org/10.1214/EJP.v19-2725

Information

Accepted: 14 August 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1300.05279
MathSciNet: MR3248198
Digital Object Identifier: 10.1214/EJP.v19-2725

Subjects:
Primary: 05C80
Secondary: 05C20‎, 05C38, 60C05

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