Geometrical quantity on random checkerboards on the regular torus

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 $\varepsilon > 0$. In this article, we establish a probabilistic property of the functional $\ell^T$: the random law $\ell^T\left(\omega_\varepsilon^n\right)$ converges in probability to $\varepsilon$ as $n \to + \infty$.Considering random subsets $\omega_\varepsilon^n$ allows us to construct subsets $\omega$ such that $\ell^T\left(\omega\right) = |\omega|$.