## Electronic Journal of Probability

### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.ejp/1465065711

Digital Object Identifier
doi:10.1214/EJP.v19-2725

Mathematical Reviews number (MathSciNet)
MR3248198

Zentralblatt MATH identifier
1300.05279

Rights

#### Citation

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

#### References

• Alm, Sven Erick; Linusson, Svante. A counter-intuitive correlation in a random tournament. Combin. Probab. Comput. 20 (2011), no. 1, 1–9.
• Alm, Sven Erick; Janson, Svante; Linusson, Svante. Correlations for paths in random orientations of $G(n,p)$ and $G(n,m)$. Random Structures Algorithms 39 (2011), no. 4, 486–506.
• Grimmett, Geoffrey R. Infinite paths in randomly oriented lattices. Random Structures Algorithms 18 (2001), no. 3, 257–266.
• Janson, Svante; Luczak, Malwina J. Susceptibility in subcritical random graphs. J. Math. Phys. 49 (2008), no. 12, 125207, 23 pp.
• McDiarmid, Colin. General percolation and random graphs. Adv. in Appl. Probab. 13 (1981), no. 1, 40–60.
• Heinrich Weber, phLehrbuch der Algebra, Zweite Auflage, Erster Band. Friedrich Vieweg und Sohn, Braunschweig (1898).