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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Universality of the nodal length of bivariate random trigonometric polynomials


Authors: Jürgen Angst, Viet-Hung Pham and Guillaume Poly
Journal: Trans. Amer. Math. Soc. 370 (2018), 8331-8357
MSC (2010): Primary 26C10; Secondary 30C15, 42A05, 60F17, 60G55
DOI: https://doi.org/10.1090/tran/7255
Published electronically: July 12, 2018
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Abstract: We consider random trigonometric polynomials of the form

$\displaystyle f_n(x,y)=\sum _{1\le k,l \le n} a_{k,l} \cos (kx) \cos (ly), $

where the entries $ (a_{k,l})_{k,l\ge 1}$ are i.i.d. random variables that are centered with unit variance. We investigate the length $ \ell _K(f_n)$ of the nodal set $ Z_K(f_n)$ of the zeros of $ f_n$ that belong to a compact set $ K \subset \mathbb{R}^2$. We first establish a local universality result, namely we prove that, as $ n$ goes to infinity, the sequence of random variables $ n\, \ell _{K/n}(f_n)$ converges in distribution to a universal limit which does not depend on the particular law of the entries. We then show that at a macroscopic scale, the expectation of $ \ell _{[0,\pi ]^2}(f_n)/n$ also converges to an universal limit. Our approach provides two main byproducts: (i) a general result regarding the continuity of the volume of the nodal sets with respect to $ C^1$-convergence which refines previous findings of Rusakov and Selezniev, Iksanov, Kabluchko, and Marynuch, and Azaís, Dalmao, León, Nourdin, and Poly, and (ii) a new strategy for proving small ball estimates in random trigonometric models, providing in turn uniform local controls of the nodal volumes.

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Additional Information

Jürgen Angst
Affiliation: IRMAR, University of Rennes 1, Rennes, France
Email: jurgen.angst@univ-rennes1.fr

Viet-Hung Pham
Affiliation: Vietnamese Institute for Advanced Study in Mathematics, Ha Noi, Viet Nam
Email: pgviethung@gmail.com

Guillaume Poly
Affiliation: IRMAR, University of Rennes 1, Rennes, France
Email: guillaume.poly@univ-rennes1.fr

DOI: https://doi.org/10.1090/tran/7255
Received by editor(s): October 28, 2016
Received by editor(s) in revised form: January 30, 2017, and March 10, 2017
Published electronically: July 12, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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