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

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