Critical sets of random smooth functions on products of spheres

Abstract

We prove a Chern-Lashof type formula computing the expected number of critical points of smooth function on a smooth manifold $M$ randomly chosen from a finite dimensional subspace $V\subset C^\infty(M)$ equipped with a Gaussian probability measure. We then use this formula this formula to find the asymptotics of the expected number of critical points of a random linear combination of a large number eigenfunctions of the Laplacian on the round sphere, tori, or a products of two round spheres. In the case $M=S^1$ we show that the number of critical points of a trigonometric polynomial of degree $\leq ν$ is a random variable $Z_ν$ with expectation $E(Z_ν)\sim 2\sqrt{0.6}\,ν$ and variance $var(Z_ν)\sim cν$ as $ν\to \infty$, $c\approx 0.35$.