Build a random tournament $T=(V,E)$ on $V=\{1,\ldots, n\}$ in the following fashion: for $i < j\in \{1,\ldots, n\}$ let the probability be $0.5$ whether $(i,j)\in E$ or $(j,i)\in E$ (in a tournament, exactly one of these is the case).


Let $E(n)$ be the expected value of the longest directed circuit in $T$.


Is there a formula for $E(n)$? If not: do we know anything about $\lim\_{n\to\infty}\frac{E(n)}{\sqrt{n}}$ or $\lim\_{n\to\infty}\frac{E(n)}{\log(n)}$?



Length of longest directed circuit in random tournament
A random tournament is strongly connected with probability tending to 1 exponentially fast, and all strongly connected tournaments have hamiltonian cycles.


