Correlations and screening of topological charges in Gaussian random fields

Two-point topological charge correlation functions of several types of geometric singularity in Gaussian random fields are calculated explicitly, using a general scheme: zeros of n-dimensional random vectors, signed by the sign of their Jacobian determinant; critical points (gradient zeros) of real scalars in two dimensions signed by the Hessian; and umbilic points of real scalars in two dimensions, signed by their index. The functions in each case depend on the underlying spatial correlation function of the field. These topological charge correlation functions are found to obey the first Stillinger-Lovett sum rule for ionic fluids.