Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $$ \zeta_t(\theta)= \theta + t \tan (\theta + t
\tan (\theta + t \tan (\theta +\ldots))), \qquad \mathrm{Im}\, \theta >0, \;\;
t>0. $$

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The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: q-bio