We consider random trigonometric polynomials of the form f n (x, y) = 1≤k,l≤n a k,l cos(kx) cos(ly), where the entries (a k,l ) k,l≥1 are i.i.d. random variables that are centered with unit variance. We investigate the length K (f n ) of the nodal set Z K (f n ) of the zeros of f n that belong to a compact set K ⊂ R 2 . We first establish a local universality result, namely we prove that, as n goes to infinity, the sequence of random variables n 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 [0,π] 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 [RS01, IK16, ADL+ 15] 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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In this paper we consider the distribution of the maximum of a Gaussian field defined on non locally convex sets. Adler and Taylor or Azaïs and Wschebor give the expansions in the locally convex case. The present paper generalizes their results to the non locally convex case by giving a full expansion in dimension 2 and some generalizations in higher dimension. For a given class of sets, a Steiner formula is established and the correspondence between this formula and the tail of the maximum is proved. The main tool is a recent result of Azaïs and Wschebor that shows that under some conditions the excursion set is close to a ball with a random radius. Examples are given in dimension 2 and higher.

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