Critical sets of random smooth functions on compact manifolds

Given a compact, $m$-dimensional Riemann manifold $(M,g)$ and a large positive constant $L$ we denote by $U_L$ the subspace of $C^{\mathrm{\infty }}(M)$ spanned by the eigenfunctions of the Laplacian corresponding to eigenvalues $\le L$. We equip $U_L$ with the standard Gaussian probability measure induced by the $L^2$-metric on $U_L$, and we denote by $N_L$ the expected number of critical points of a random function in $U_L$. We prove that $N_L\sim C_m\dim U_L$ as $L\to \mathrm{\infty }$, where $C_m$ is an explicit positive constant that depends only on the dimension $m$ and satisfying the asymptotic estimate $\log C_m\sim\frac{m}{2}\log m$as$m\to \infty$.