Zeros and critical points of Gaussian fields: cumulants asymptotics and limit theorems

Let $f:\mathbb{R}^d \to \mathbb{R}^k$ be a smooth centered stationary Gaussian field and $\mathcal{B} \subset \mathbb{R}^d$ be a bounded Borel set. In this paper, we determine the asymptotics as $R \to \infty$ of all the cumulants of the $(d-k)$-dimensional volume of $f^{-1}(0) \cap R\mathcal{B}$. When $k=1$, we obtain similar asymptotics for the number of critical points of $f$ in $R\mathcal{B}$. Our main hypotheses are some regularity and non-degeneracy of the field, as well as mild integrability conditions on the first derivatives of its covariance kernel. As corollaries of these cumulants estimates, we deduce a strong Law of Large Numbers and a Central Limit Theorem for the nodal volume (resp.~the number of critical points) of a regular and non-degenerate enough field whose covariance decays fast enough at infinity. Our results hold more generally for a one-parameter family $(f_R)$ of Gaussian fields admitting a stationary local scaling limit as $R \to \infty$, for example Kostlan polynomials in the large degree limit. They also hold for the random measures of integration over the vanishing locus of $f_R$ as $R \to +\infty$.