A stochastic Gauss-Bonnet-Chern formula

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem stating that if the bundle and the manifold are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble.