Electronic Journal of Probability

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

Sven Erick Alm, Svante Janson, and Svante Linusson

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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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 69, 14 pp.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 05C20: Directed graphs (digraphs), tournaments 05C38: Paths and cycles [See also 90B10] 60C05: Combinatorial probability

Random directed graph correlation directed paths

This work is licensed under a Creative Commons Attribution 3.0 License.


Alm, Sven Erick; Janson, Svante; Linusson, Svante. First critical probability for a problem on random orientations in $G(n,p)$. Electron. J. Probab. 19 (2014), paper no. 69, 14 pp. doi:10.1214/EJP.v19-2725. https://projecteuclid.org/euclid.ejp/1465065711

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