Cycle structure of random permutations with cycle weights

TL;DR

This paper studies the properties of cycle lengths and counts in weighted random permutations, revealing how parameters influence typical cycle structures and showing finite cycles follow Poisson distributions.

Contribution

It provides a detailed analysis of cycle structures in weighted permutations, highlighting the dependence on parameters and the Poisson nature of finite cycles.

Findings

Typical cycle lengths depend on parameters

Total number of cycles varies with parameters

Finite cycles are often Poisson distributed

Abstract

We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the parameters, while the distributions of finite cycles are usually independent Poisson random variables.