Distribution of the roots of Eulerian polynomials
Paul Melotti
TL;DR
The paper provides a new combinatorial proof that the zeros of Eulerian polynomials induce an empirical measure converging to a log-Cauchy distribution. By studying the transformed roots $u_{n,k}=\frac{1}{1-x_{n,k}}$ and computing their moments exactly, it shows $\frac{1}{n+1}\sum_{k=1}^n u_{n,k}^p = \frac{C_p}{p!}$ for all $1\le p\le n$, where $C_p$ are Cauchy numbers of the second kind, enabling a concrete weak limit via the method of moments. The limiting measure $\nu$ on $[0,1]$ has moments $\frac{C_p}{p!}$ and a Stieltjes transform $S_\nu(t) = \frac{1}{t(t-1)\log(1-1/t)}$, and its pushforward under $u \mapsto \frac{1}{u}-1$ yields $\mu$ with density $\frac{1}{t(\pi^2+\log^2 t)}$ on $(0,\infty)$, equivalently the distribution of $e^{\pi Z}$ for $Z$ standard Cauchy. This extends the probabilistic understanding of Eulerian root distributions and highlights a precise combinatorial mechanism behind the limiting law.
Abstract
We give a new proof that the empirical measures of the roots of Eulerian polynomials converge to a certain log-Cauchy distribution. To do so, we show that each moment of the roots of a related family of polynomials not only converge, but in fact become ultimately constant. These asymptotic moments are expressed in terms of Cauchy numbers of the second kind.