The number of critical points of a Gaussian field: finiteness of moments (Q6632867)

Let \(F:\mathcal{U}\to \mathbb{R}^d\) be a Gaussian random field on an open domain \(\mathcal{U}\) in \(\mathbb{R}^d\) with almost surely \(\mathcal{C}^p\) paths, where \(p\in\mathbb{N}\). The authors prove that, if for every \(x\in\mathcal{U}\), the random Gaussian vector composed of all mixed partial derivatives of \(F\) up to the order \(p-1\) is non-degenerate, then, for every compact subset \(K\) of \(\mathcal{U}\), the cardinality of the set \(\{x\in K: F(x)=0\}\) has finite \(p\)th moment. In the previous works, this has been established only for moments of order lower than three.\N\NSimilar statements are obtained for holomorphic random fields on a subset of \(\mathbb{C}^d\) with values in \(\mathbb{C}^d\) and random fields which are given by the gradient of a real-valued Gaussian function \(f\) on \(\mathcal{U}\), and so for the number of the critical points of \(f\). In particular, the authors show the finiteness of moments of the nodal volumes and the number of critical points of a large class of smooth, or holomorphic, Gaussian fields, including the Bargmann-Fock ensemble.