Asymptotic correlations of metrics on the symmetric groups

TL;DR

This paper investigates the asymptotic joint distributions of various well-known metrics on symmetric groups, introducing a new limit metric and analyzing their asymptotic independence and distributional properties.

Contribution

It introduces a natural limit of Spearman's footrule, studies its asymptotic distribution, and extends previous work on asymptotic independence of metrics on symmetric groups.

Findings

Established asymptotic independence of several metrics on $S_n$

Introduced and analyzed the limit metric $ ho_ $ for Spearman's footrule

Provided near-optimal error estimates for asymptotic independence

Abstract

We consider the asymptotic joint distributions among several families of well-known metrics on $S_n$, the symmetric group. These include the bi-invariant metrics such as the Cayley and Hamming distance, and the left-invariant metrics such as Spearman's footrule, Kendall's tau, and the Ulam distance. We also introduce a natural limit of the Spearman family, $\rho_\infty$, and study its asymptotic distribution and relation with other metrics. This is a continuation of earlier work on the asymptotic independence of bi-invariant metrics on both $S_n$ and general linear groups over a finite field. The technique is based on some simple observation about the record map and Hammersley's device. In several cases, we give near-optimal estimate of the error term for asymptotic independence. This simplifies significantly the proof of a central limit theorem by Bai, Chao, and Liang regarding the oscillation of a permutation.