Repeated differentiation and free unitary Poisson process
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arXiv
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| Format: | Preprint |
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2021
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| author | Kabluchko, Zakhar |
| author_facet | Kabluchko, Zakhar |
| contents | 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 $ν$ 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 $ν$ 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 $(\sin \frac θ2)^{2d}$ (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 $ζ=ζ_t(θ)$ which solves the implicit equation $ζ- t \tan ζ= θ$ and satisfies $$ ζ_t(θ)= θ+ t \tan (θ+ t \tan (θ+ t \tan (θ+\ldots))), \qquad \mathrm{Im}\, θ>0, \;\; t>0. $$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_14729 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Repeated differentiation and free unitary Poisson process Kabluchko, Zakhar Probability Analysis of PDEs Classical Analysis and ODEs Complex Variables Primary: 30C15, Secondary: 30C10, 26C10, 35A25, 60B10, 60B20, 82C70, 46L54, 44A15, 30F99 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 $ν$ 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 $ν$ 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 $(\sin \frac θ2)^{2d}$ (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 $ζ=ζ_t(θ)$ which solves the implicit equation $ζ- t \tan ζ= θ$ and satisfies $$ ζ_t(θ)= θ+ t \tan (θ+ t \tan (θ+ t \tan (θ+\ldots))), \qquad \mathrm{Im}\, θ>0, \;\; t>0. $$ |
| title | Repeated differentiation and free unitary Poisson process |
| topic | Probability Analysis of PDEs Classical Analysis and ODEs Complex Variables Primary: 30C15, Secondary: 30C10, 26C10, 35A25, 60B10, 60B20, 82C70, 46L54, 44A15, 30F99 |
| url | https://arxiv.org/abs/2112.14729 |