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 d trigonometric polynomial are distributed according to some probability measure ν ν in the large d d limit, then the zeroes of its [ 2 t d ] [2td] -th derivative, where t > 0 t>0 is fixed, are distributed according to the free multiplicative convolution of ν ν and the free unitary Poisson distribution with parameter t t . In the simplest special case, our result states that the zeroes of the [ 2 t d ] [2td] -th derivative of the trigonometric polynomial ( sin ⁡ θ 2 ) 2 d (sin θ 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 t , in the large d d limit. The latter distribution is defined in terms of the function ζ = ζ t ( θ ) ζ =ζ _t(θ ) which solves the implicit equation ζ − t tan ⁡ ζ = θ ζ - t tan ζ = θ and satisfies ζ t ( θ ) = θ + t tan ⁡ ( θ + t tan ⁡ ( θ + t tan ⁡ ( θ + … ) ) ) , Im ⁡ θ > 0 , t > 0. {equation*} ζ _t(θ )= θ + t tan (θ + t tan (θ + t tan (θ +… ))), Im θ >0, \ t>0. {equation*}