Sawtooth models and asymptotic independence in large compositions

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 $+\infty$.