  We study the local repulsion between critical points of a stationary
isotropic smooth planar Gaussian field. We show that the critical points can
experience a soft repulsion which is maximal in the case of the random planar
wave model, or a soft attraction of arbitrary high order. If the type of
critical points is specified (extremum, saddle point), the points experience a
hard local repulsion, that we quantify with the precise magnitude of the second
factorial moment of the number of points in a small ball.
