Random permutations without macroscopic cycles

@article{Betz2020RandomPW,
  title={Random permutations without macroscopic cycles},
  author={Volker Betz and Helge Schafer and Dirk Zeindler},
  journal={The Annals of Applied Probability},
  year={2020}
}
We consider uniform random permutations of length n conditioned to have no cycle longer than nβ with 0 < β < 1, in the limit of large n. Since in unconstrained uniform random permutations most of the indices are in cycles of macroscopic length, this is a singular conditioning in the limit. Nevertheless, we obtain a fairly complete picture about the cycle number distribution at various lengths. Depending on the scale at which cycle numbers are studied, our results include Poisson convergence, a… 
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