  Random plane wave is conjectured to be a universal model for high-energy
eigenfunctions of the Laplace operator on generic compact Riemanian manifolds.
This is known to be true on average. In the present paper we discuss one of
important geometric observable: critical points. We first compute one-point
function for the critical point process, in particular we compute the expected
number of critical points inside any open set. After that we compute the
short-range asymptotic behaviour of the two-point function. This gives an
unexpected result that the second factorial moment of the number of critical
points in a small disc scales as the fourth power of the radius.
