Résumé en anglais

This paper is concerned with the properties of Gaussian random fields defined on a riemannian homogeneous space, under the assumption that the probability distribution be invariant under the isometry group of the space. We first indicate, building on early results of Yaglom, how the available information on group-representation-theory-related special functions makes it possible to give completely explicit descriptions of these fields in many cases of interest. We then turn to the expected size of the zero-set: extending two-dimensional results from Optics and Neuroscience, we show that every invariant field comes with a natural unit of volume (defined in terms of the geometrical redundancies in the field) with respect to which the average size of the zero-set depends only on the dimension of the source and target spaces, and not on the precise symmetry exhibited by the field. Both the volume unit and the associated density of zeroes can in principle be evaluated from a single sample of the field, and our result provides a numerical signature for the fact that a given individual map be a sample from an invariant Gaussian field.