On the flat torus $\mathbb{ T}^m=\mathbb{ R}^m/\mathbb{ Z}^m$ with angular coordinates $\vec{ \theta}$ we consider the random function $F_R=\mathfrak{ a}\big(\, R^{ -1} \sqrt{ \Delta}\,\big) W$, where $R>0$, $\Delta$ is the Laplacian on this flat torus, $\mathfrak{ a}$ is an even Schwartz function on $\mathbb{ R}$ such that $\mathfrak{ a}(0)>0$ and $W$ is the Gaussian white noise on $\mathbb{ T}^m$ viewed as a random generalized function. For any $f\in C(\mathbb{ T}^m)$ we set \[ Z_R(f):=\sum_{ \nabla F_R(\vec{ \theta})=0} f(\vec{ \theta}) \] We prove that if the support of $f$ is contained in a geodesic ball of $\mathbb{ T}^m$, then the variance of $Z_R(f)$ is asymptotic to $const\times R^{ m}$ as $R\to\infty$. We use this to prove that if $m\geq 2$, then as $N\to\infty$ the random measures $N^{ -m}Z_N(-)$ converge a.s. to an explicit multiple of the volume measure on the flat torus.
Categories: math.PR