Repeated differentiation and free unitary Poisson process
We investigate the hydrodynamic behaviour 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, this result states that the zeroes of the [2td]-th derivative of the trigonometric polynomial (sin [ theta/ 2])^{2d} are distributed according to the free unitary Poisson distribution with parameter t, in the large d limit. The latter distribution can be 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 +…))). The talk is based on the preprint https://arxiv.org/abs/2112.14729