Summary: We investigate the number of critical points of a Gaussian random smooth function \(u^{\epsilon}\) on the \(m\)-torus \(T^{m} := \mathbb{R}^{m}/\mathbb{Z}^{m}\) approximating the Gaussian white noise as \(\epsilon \to 0\). Let \(N(u^{\epsilon})\) denote the number of critical points of \(u^{\epsilon}\). We prove the existence of constants \(C,\; C^{\prime}\) such that as \(\epsilon\) goes to zero, the expectation of the random variable \(\epsilon^{m}N(u^{\epsilon})\) converges to \(C\), while its variance is extremely small and behaves like \(C^{\prime}\epsilon^{m}\).{
©2016 American Institute of Physics}