A generalization of the Erd\H{o}s-Tur\'an law for the order of random permutation

TL;DR

This paper extends the Erdős-Turán law to a broader class of random permutations derived from stick-breaking processes, establishing a central limit theorem for their order's logarithm.

Contribution

It introduces a new framework linking stick-breaking partitions with permutation cycle structures and proves a CLT for the permutation order's logarithm under these models.

Findings

Proves a CLT for the logarithm of permutation order.

Extends Erdős-Turán law beyond uniform and Ewens' permutations.

Uses perturbed random walks to analyze cycle length sums.

Abstract

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on $n$ integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erd\H{o}s-Tur\'an law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM$(\theta)$ distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.