Repeated differentiation and free unitary Poisson process

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 $$ \zeta_t(\theta)= \theta + t \tan (\theta + t \tan (\theta + t \tan (\theta +\ldots))), \qquad \mathrm{Im}\, \theta >0, \;\; t>0. $$