Random permutations without macroscopic cycles

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 central limit theorem, a shape theorem and two different functional central limit theorems.