Total variation distance and the Erd\H{o}s-Tur\'an law for random permutations with polynomially growing cycle weights

TL;DR

This paper analyzes random permutations with polynomial cycle weights, establishing convergence of cycle counts to independent Poissons, a CLT for permutation order, and a Brownian motion limit for small cycles, extending classical laws.

Contribution

It introduces saddle-point analysis for this permutation model, proving convergence results and extending the Erdős-Turán Law to polynomially weighted permutations.

Findings

Total variation distance converges to 0 under certain conditions.

Central limit theorem for permutation order established.

Brownian motion limit for small cycles demonstrated.

Abstract

We study the model of random permutations of $n$ objects with polynomially growing cycle weights, which was recently considered by Ercolani and Ueltschi, among others. Using saddle-point analysis, we prove that the total variation distance between the process which counts the cycles of size $1, 2, ..., b$ and a process $(Z_1, Z_2, ..., Z_b)$ of independent Poisson random variables converges to $0$ if and only if $b=o(\ell)$ where $\ell$ denotes the length of a typical cycle in this model. By means of this result, we prove a central limit theorem for the order of a permutation and thus extend the Erd\H{o}s-Tur\'an Law to this measure. Furthermore, we prove a Brownian motion limit theorem for the small cycles.