Short cycles of random permutations with cycle weights: point processes approach

TL;DR

This paper analyzes the asymptotic distribution of short cycles in random permutations with cycle weights using a point process approach, demonstrating convergence to a Poisson process and deriving related limit theorems.

Contribution

It introduces a point process framework for studying cycle structures in weighted permutations and proves convergence results for a broad class of cycle weights.

Findings

Cycle point processes converge to Poisson processes as permutation size grows.

The approach yields new limit theorems for cycle statistics.

Results apply to a wide range of cycle weight functions.

Abstract

We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on $\{1,\ldots,n\}$. The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as $n\to\infty$ for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles.