## Universality of the nodal length of bivariate random trigonometric polynomials

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- by Jürgen Angst, Viet-Hung Pham and Guillaume Poly PDF
- Trans. Amer. Math. Soc.
**370**(2018), 8331-8357 Request permission

## Abstract:

We consider random trigonometric polynomials of the form \[ f_n(x,y)=\sum _{1\le k,l \le n} a_{k,l} \cos (kx) \cos (ly), \] 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.## References

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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
- MR Author ID: 1027015
- Email: pgviethung@gmail.com
**Guillaume Poly**- Affiliation: IRMAR, University of Rennes 1, Rennes, France
- MR Author ID: 997488
- Email: guillaume.poly@univ-rennes1.fr
- 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
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3864378