The order of large random permutations with cycle weights

TL;DR

This paper extends classical results on the order of random permutations to those with cycle weights, establishing local limit theorems and large deviations estimates, including for permutations with polynomial cycle weights.

Contribution

It introduces new local limit theorems and large deviations estimates for permutation orders under generalized Ewens measures and polynomial cycle weights.

Findings

Established a local limit theorem for permutation order.

Derived large deviations estimates under additional moment conditions.

Extended results to permutations with polynomial cycle weights.

Abstract

The order $O_n(\sigma)$ of a permutation $\sigma$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $\sigma$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erd\"os and Tur\'an who proved in 1965 that $\log O_n$ satisfies a central limit theorem. We extend this result to the so-called \textit{generalized Ewens measure} in a previous paper. In this paper, we establish a local limit theorem as well as, under some extra moment condition, a precise large deviations estimate. These properties are new even for the uniform measure. Furthermore, we provide precise large deviations estimates for random permutations with polynomial cycle weights.