5 problems

A spatial random permutation (SRP) is a random bijection of a finite lattice or set of sites; the Mallows model is one SRP whose graph has a concentrated band structure around the…

Fyodorov–Muirhead conjecture. The expected diameter of the cycle satisfies

Emergence of macroscopic cycles conjecture. The longest-cycle length has the asymptotic scales above, with macroscopic cycles in the regimes where $C_\beta^N=\Theta(N)$. The predic…

Linear box-counting dimension conjecture. The function $d(\alpha)$ is a linear function of $\alpha$ for small $\alpha$. The authors explicitly state that their numerical accuracy i…

A spatial random permutation (SRP) is sampled in the scaling limit, and consider its long cycles and their box-counting dimension, obtained from the scaling of the number of boxes…