Sawtooth models and asymptotic independence in large compositions

TL;DR

This paper introduces a simplified probabilistic model for compositions, leading to new insights on the asymptotic independence of permutation elements with fixed descent sets.

Contribution

It presents a more general probabilistic framework that improves upon previous models for compositions and analyzes the asymptotic independence in permutations.

Findings

Permutation elements become independent as their indices grow apart.

New estimates on the behavior of permutations with fixed descent sets.

The model simplifies analysis while extending previous results.

Abstract

In this paper we improve the probabilistic approach to compositions of Ehrenborg, Levin and Readdy by introducing a simpler but more general probabilistic model. As consequence we get some new estimates on the behavior of a uniform random permutation $\sigma$ having a fixed descent set. In particular we show that independently of the shape of the descent set, $\sigma(i)$ and $\sigma(j)$ become independent when $i-j$ tends to $+\infty$.