Geometrical quantity on random checkerboards on the regular torus

Institut Denis Poisson, UMR-CNRS 7013, Université de Tours, France

^* Corresponding author: lea-gohier@hotmail.com

Received: 28 August 2024
Accepted: 11 April 2025

Abstract

In the study of the observability of the wave equation (here on (0, T) × T^d, where T^d is the d-dimensional torus), a condition naturally emerges as a sufficient observability condition. This condition, which writes ℓ^T (ω) > 0, signifies that the smallest time spent by a geodesic in the subset ω ⊂ T^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 n^d, n ∈ 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 ℓ^T : the random law ℓ^T (ωⁿ_ε) converges in probability to ε as n → +∞.