Repeated differentiation and free unitary Poisson process

TL;DR

This paper studies how the zeroes of repeated derivatives of trigonometric polynomials distribute in the large degree limit, revealing a connection to free probability and the free unitary Poisson distribution.

Contribution

It establishes a link between the zeroes of derivatives of trigonometric polynomials and free multiplicative convolution with the free unitary Poisson distribution, extending understanding of their asymptotic behavior.

Findings

Zeroes of derivatives follow free multiplicative convolution distribution

Distribution converges to free unitary Poisson in large degree limit

Explicit implicit equation characterizes the distribution's behavior

Abstract

We investigate the hydrodynamic behavior of zeroes of trigonometric polynomials under repeated differentiation. We show that if the zeroes of a real-rooted, degree $d$ trigonometric polynomial are distributed according to some probability measure $\nu$ in the large $d$ limit, then the zeroes of its $[2td]$-th derivative, where $t>0$ is fixed, are distributed according to the free multiplicative convolution of $\nu$ and the free unitary Poisson distribution with parameter $t$. In the simplest special case, our result states that the zeroes of the $[2td]$-th derivative of the trigonometric polynomial $(\sin \frac \theta 2)^{2d}$ (which can be thought of as the trigonometric analogue of the Laguerre polynomials) are distributed according to the free unitary Poisson distribution with parameter $t$, in the large $d$ limit. The latter distribution is defined in terms of the function $\zeta=\zeta_t(\theta)$ which solves the implicit equation $\zeta - t \tan \zeta = \theta$ and satisfies $$ \zeta_t(\theta)= \theta + t \tan (\theta + t \tan (\theta + t \tan (\theta +\ldots))), \qquad \mathrm{Im}\, \theta >0, \;\; t>0. $$