# QuickSort: Improved right-tail asymptotics for the limiting distribution, and large deviations

## TL;DR

This paper refines the asymptotic bounds on the right tail of the QuickSort distribution, providing more precise large deviation results for the number of comparisons, improving upon previous bounds and conjectures.

## Contribution

It offers significantly improved asymptotic bounds on the right tail of the QuickSort limiting distribution and its density, matching conjectured asymptotics and refining large deviation estimates.

## Key findings

- Established upper bounds matching conjectured asymptotics
- Derived sharper large deviation results for QuickSort comparisons
- Improved understanding of the distribution's tail behavior

## Abstract

We substantially refine asymptotic logarithmic upper bounds produced by Svante Janson (2015) on the right tail of the limiting QuickSort distribution function $F$ and by Fill and Hung (2018) on the right tails of the corresponding density $f$ and of the absolute derivatives of $f$ of each order. For example, we establish an upper bound on $\log[1 - F(x)]$ that matches conjectured asymptotics of Knessl and Szpankowski (1999) through terms of order $(\log x)^2$; the corresponding order for the Janson (2015) bound is the lead order, $x \log x$. Using the refined asymptotic bounds on $F$, we derive right-tail large deviation (LD) results for the distribution of the number of comparisons required by QuickSort that substantially sharpen the two-sided LD results of McDiarmid and Hayward (1996).