A central limit theorem for random walks in random labyrinths

Abstract

A beam of light shines through the lattice Z d , and is subjected to reflections determined by a random environment of mirrors at the vertices of Z d . The behaviour of the light ray is investigated under the hypothesis that the environment contains a strictly positive density of vertices at which the light behaves in the manner of a random walk. When d≥2 and the density of non-trivial reflectors is sufficiently small, the environment contains almost surely a unique infinite `inter-illuminating' class of vertices. Furthermore, when the light beam originates within this class, then its trajectory obeys a functional central limit theorem with a strictly positive diffusion constant. These facts are obtained using percolation-type arguments, together with the invariance principle proposed by Kipnis and Varadhan.