Roots of Eulerian polynomials, old and new

Let's count permutations by their number of descents, and we get the sequence of Eulerian polynomials. If we plot these, we see that they have only real roots, a result proven by Frobenius in 1910. Taking a probabilist viewpoint, we may then wonder about the limiting distribution of the empirical measures of these roots. The asymptotic measure is a surprising log-Cauchy distribution, which was proven by Sobolev in 1977. I will present another proof, that gives not only the asymptotic but also the exact distribution at every step.