Correlations for paths in random orientations of G(n,p) and G(n,m)


Sven Erick Alm, Svante Janson and Svante Linusson

Uppsala University, Uppsala University and Royal Institute of Technology


In Random Structures and Algorithms Vol. 39, Issue 4, 486-506 (2011).


We study random graphs, both $G(n,p)$ and $G(n,m)$, with random orientations on the edges. For three fixed distinct vertices $s,a,b$ we study the correlation, in the combined probability space, of the events $\{a\to s\}$ and $\{s\to b\}$.
For $\gnp$, we prove that there is a $\pc=1/2$ such that for a fixed $p<\pc$ the correlation is negative for large enough $n$ and for $p>\pc$ the correlation is positive for large enough $n$. %For $G(n,p)$ it is proved that $\pc=1/2$ and We conjecture that for a fixed $n\ge 27$ the correlation changes sign three times for three critical values of $p$.
For $G(n,m)$ it is similarly proved that, with $p=m/\binom{n}{2}$, there is a critical $\pc$ that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any $\ell$ directed edges in $\gnm$, is thought to be of independent interest.
We present exact recursions to compute $\P(a\to s)$ and $\P(a\to s, s\to b)$. We also briefly discuss the corresponding question in the quenched version of the problem.

2014-08-18, Sven Erick Alm,