Appears in collection : Random walks: applications and interactions / Marches aléatoires: applications et interactions

We recall the tubular and planar Lorentz gas model, which can be viewed as a deterministic version of the random walk on $\mathbb{Z}^d, d=1$ or $d=2$. The associated displacement function obeys the same laws as a random walk on $\mathbb{Z}^d$ with finite variance (finite horizon) or infinite variance (infinite horizon). In the infinite variance case, the first moment of the displacement function 'barely' fails to be $L^2$. In both cases, finite and infinite variance, the asymptotic limit law of the displacement function is Gaussian (just different scaling). I will recall several stochastic properties of the displacement function and enquire about the possibility of breaking the Gaussian law.