Geometrical quantity on random checkerboards on the regular torus

Abstract

In the study of the observability of the wave equation (here on $(0,T)\times {\mathbb{T}}^d$, where ${\mathbb{T}}^d$ is the d-dimensional torus), a condition naturally emerges as a sufficient observability condition. This condition, which writes ${\ell }^T\left(\omega \right)>0$, signifies that the smallest time spent by a geodesic in the subset $\omega \subset {\mathbb{T}}^d$ during time $T$ is non-zero. In other words, the subset $\omega$ detects any geodesic propagating on the d-dimensional torus during time $T$. Here, the subset $\omega$ is randomly defined by drawing a grid of $n^d$, $n\in \mathbb{N}$, small cubes of equal size and by adding them to $\omega$ with probability $\epsilon >0$. In this article, we establish a probabilistic property of the functional ${\ell }^T$: the random law ${\ell }^T\left({\omega }_{\epsilon }^n\right)$ converges in probability to $\epsilon$ as $n\to +\infty$.Considering random subsets ${\omega }_{\epsilon }^n$ allows us to construct subsets $\omega$ such that ${\ell }^T\left(\omega \right)=|\omega |$.