A law of large numbers concerning the number of critical points of isotropic Gaussian functions

Abstract

We investigate the distribution of critical points of certain isotropic random functions $\Phi$ on ${\mathbb{R}}^m$. We show that the distribution of critical points of $\Phi (Rx)$, suitably normalized, converge a.s. and $L^2$ as random measures to the (deterministic) Lebesgue measure as $R\to \infty$. We achieve this by producing precise asymptotics of the second moments of these distributions as $R\to \infty$.

19 pages, simplified proofs, strengthened the main result