Gaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse... Show moreGaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse phenomena, ranging from structure formation in condensed matter and cosmology to biomedical imaging. The standard test of non-Gaussianity is to measure higher-order correlation functions. This work takes a different route. It is shown how geometric and topological properties of Gaussian fields, in particular the statistics of extrema and umbilical points, are modified by the presence of a non-Gaussian perturbation. The resulting discrepancies give an independent way to detect and quantify non-Gaussianities. Both local and nonlocal mechanisms that generate non-Gaussian fields are considered, both statically and dynamically through nonlinear diffusion. The effects of coarse-graining are also investigated. Show less
