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 $R\to\infty$.