Long cycle of random permutations with polynomially growing cycle weights

TL;DR

This paper investigates the asymptotic properties of long cycles in random permutations with polynomially increasing cycle weights, revealing their convergence to independent exponential-like distributions using advanced analytical methods.

Contribution

It introduces a novel analysis of long cycle behavior in permutations with polynomial weights, employing generating functions and saddle point techniques.

Findings

Longest cycle length converges after normalization

Cycle length differences become independent and exponentially distributed

Methodology applies saddle point analysis to permutation cycles

Abstract

We study the asymptotic behavior of the long cycles of a random permutation of $n$ objects with respect to multiplicative measures with polynomial growing cycle weights. We show that the longest cycle and the length differences between the longest cycles converge, after suitable normalisation, in distribution to iid random variables $(Z_j)_{j\in\mathbb{N}}$ such that $\exp(-Z_j)$ is exponentially distributed. Our method is based on generating functions and the saddle point method.