Random permutations without macroscopic cycles

TL;DR

This paper investigates the cycle structure of large uniform random permutations conditioned to have no cycles longer than a certain power of n, revealing diverse probabilistic behaviors depending on the cycle length scale.

Contribution

It provides a comprehensive analysis of cycle distributions in constrained permutations, including convergence results and limit theorems, extending understanding of permutation cycle structures under singular conditioning.

Findings

Poisson convergence of cycle counts at certain scales

Central limit theorem for cycle number fluctuations

Shape theorem and functional central limit theorems

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.