Critical Points of Multidimensional Random Fourier Series: Central Limits

We investigate certain families X~, 0 < ~ 1 of stationary Gaussian random smooth functions on the m-dimensional torus T := R/Z approaching the white noise as ~ → 0. We show that there exists universal constants c1, c2 > 0 such that for any cube B ⊂ R of size r ≤ 1/2, the number of critical points of X~ in the region B mod Z ⊂ T has mean ∼ c1 vol(B)~−m, variance ∼ c2 vol(B)~−m, and satisfies a central limit theorem as ~↘ 0.