Universality of the nodal length of bivariate random trigonometric polynomials

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.