Sharp large deviations and concentration inequalities for the number of descents in a random permutation

TL;DR

This paper establishes sharp large deviation principles and precise concentration inequalities for the number of descents in a random permutation, advancing the understanding of its probabilistic behavior through martingale and distribution-based methods.

Contribution

It introduces two novel approaches to analyze the number of descents, providing the first sharp large deviation and concentration results for this permutation statistic.

Findings

Number of descents satisfies a sharp large deviation principle.

A precise concentration inequality involving the rate function is derived.

The methods include martingale decomposition and Irwin-Hall distribution analysis.

Abstract

The goal of this paper is to go further in the analysis of the behavior of the number of descents in a random permutation. Via two different approaches relying on a suitable martingale decomposition or on the Irwin-Hall distribution, we prove that the number of descents satisfies a sharp large deviation principle. A very precise concentration inequality involving the rate function in the large deviation principle is also provided.