Random permutations without macroscopic cycles

Abstract

We consider uniform random permutations of length $n$ conditioned to have no cycle longer than $n^{\beta }$ with $0<\beta <1$, 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.