The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles

TL;DR

This paper studies the asymptotic behavior of cycle structures in weighted random permutations, extending classical models and establishing convergence results, including Poisson and mod-Poisson limits, using combinatorics and complex analysis.

Contribution

It introduces a generalized weighted measure on permutations, proves convergence to independent Poisson variables, and establishes mod-Poisson convergence with error estimates.

Findings

Cycle process converges to independent Poisson variables.

Total number of cycles satisfies a central limit theorem.

Established mod-Poisson convergence and Poisson approximation.

Abstract

The goal of this paper is to analyse the asymptotic behavior of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables and to establish a central limit theorem for the total number of cycles. Our methods allow us to obtain an asymptotic estimate of the characteristic functions of the different random vectors of interest together with an error estimate, thus having a control on the speed of convergence. In fact we are able to prove a finer convergence for the total number of cycles, namely \textit{mod-Poisson convergence}. From there we apply previous results on mod-Poisson convergence to obtain Poisson approximation for the total number of cycles as well as large deviations estimates.