3 problems

Poisson–Dirichlet conjecture. If $0<\alpha=\alpha_t<1$ is the macroscopic-cycle fraction, then

Tóth's macroscopic-cycle conjecture. For $d\geq3$ and $\beta>\beta_c$, macroscopic cycles form: roughly $\alpha L^d$ particles belong to macroscopic cycles and roughly…

Short-cycle conjecture. The permutation has only short and finite cycles.