Geometrical quantity on random checkerboards on the regular torus

In the study of the observability of the wave equation (here on (0,T)× ᵈ, where ᵈ is the d-dimensional torus), a condition naturally emerges as a sufficient observability condition. This condition, which writes T(ω)>0, signifies that the smallest time spent by a geodesic in the subset ω⊂ ᵈ during time T is non-zero. In other words, the subset ω detects any geodesic propagating on the d-dimensional torus during time T. Here, the subset ω is randomly defined by drawing a grid of nᵈ, n∈, small cubes of equal size and by adding them to ω with probability ε>0. In this article, we establish a probabilistic property of the functional T: the random law T(ω_εⁿ) converges in probability to ε as n → + ∞.