Functional central limit theorem for topological functionals of Gaussian critical points

We consider Betti numbers of the excursion of a smooth Euclidean Gaussian field restricted to a rectangular window, in the asymptotics where the window grows to Rd. With motivations coming from topological data analysis, we derive a functional central limit theorem where the varying argument is the thresholding parameter, under assumptions of regularity and covariance decay for the field and its derivatives. We also show fixed- and multi-level CLTs coming from martingale-based techniques inspired by the theory of geometric stabilisation, and limiting nondegenerate variance.