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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> trigonometric polynomial are distributed according to some probability measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"> <mml:semantics> <mml:mi> ν </mml:mi> <mml:annotation encoding="application/x-tex">\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the large <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> limit, then the zeroes of its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 2 t d right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>2</mml:mn> <mml:mi>t</mml:mi> <mml:mi>d</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[2td]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -th derivative, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is fixed, are distributed according to the free multiplicative convolution of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"> <mml:semantics> <mml:mi> ν </mml:mi> <mml:annotation encoding="application/x-tex">\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the free unitary Poisson distribution with parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In the simplest special case, our result states that the zeroes of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 2 t d right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>2</mml:mn> <mml:mi>t</mml:mi> <mml:mi>d</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[2td]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -th derivative of the trigonometric polynomial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis sine StartFraction theta Over 2 EndFraction right-parenthesis Superscript 2 d"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>sin</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mfrac> <mml:mi> θ </mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(\sin \frac \theta 2)^{2d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (which can be thought of as the trigonometric analogue of the Laguerre polynomials) are distributed according to the free unitary Poisson distribution with parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , in the large <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> limit. The latter distribution is defined in terms of the function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta equals zeta Subscript t Baseline left-parenthesis theta right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ζ </mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi> ζ </mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi> θ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\zeta =\zeta _t(\theta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which solves the implicit equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta minus t tangent zeta equals theta"> <mml:semantics> <mml:mrow> <mml:mi> ζ </mml:mi> <mml:mo> − </mml:mo> <mml:mi>t</mml:mi> <mml:mi>tan</mml:mi