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The minimum modulus of Gaussian trigonometric polynomials

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Abstract

We prove that the minimum of the modulus of a random trigonometric polynomial with Gaussian coefficients, properly normalized, has limiting exponential distribution.

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Acknowledgments

O. Y. thanks Alon Nishry for introducing him to this problem, encouraging him to work on it and for many fruitful discussions. O. Z. thanks Hoi Nguyen for suggesting this problem at an AIM meeting in August 2019, and Pavel Bleher, Nick Cook and Hoi Nguyen for stimulating discussions at that meeting concerning this problem. In particular, the realization that a linear approximation suffices when working on a net with spacing o(1/n) came out of those discussions.

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Authors and Affiliations

  1. School of Mathematics, Tel Aviv University, Ramat Aviv, 6997801, Israel

    Oren Yakir

  2. Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Ofer Zeitouni

Authors
  1. Oren Yakir
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  2. Ofer Zeitouni
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Corresponding author

Correspondence to Ofer Zeitouni.

Additional information

Dedicated to the memory of Thomas Liggett, 1944–2020

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 692452). O.Y. is supported by ISF Grants 382/15 and 1903/18.

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Yakir, O., Zeitouni, O. The minimum modulus of Gaussian trigonometric polynomials. Isr. J. Math. 245, 543–566 (2021). https://doi.org/10.1007/s11856-021-2218-x

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  • DOI: https://doi.org/10.1007/s11856-021-2218-x

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