Sharp large deviation principles for descents of random permutations.

Resumen:   A permutation $\pi_n$ of size $n$ is said to have a descent at position $k$ if $\pi_n(k) > \pi_n(k + 1)$. We define $D_n$ as the number of descents of $\pi_n$, and $D_n’$ as the number of descents of $\pi_n^{-1}$, the inverse permutation of $\pi_n$.
In this talk, I will present recent developments regarding the sharp large deviations for $D_n$ and for the pair $(D_n, D_n’)$, along with other probabilistic results when $\pi_n$ is taken uniformly at random from $S_n$, the set of permutations of size $n$. Work in collaboration with B. Bercu, M. Bonnefont, and A. Richou.