Infinite paths in a Lorentz lattice gas model

We consider infinite paths in an illumination problem on the lattice Z(2), where at each vertex, there is either a two-sided minor (with probability p greater than or equal to 0) or no mirror (with probability 1 - p). The mir rors are independently oriented NE-SW or NW-SE with equal probability. We c onsider beams of light which are shone from the origin and deflected by the mirrors. The beam of light is either periodic or unbounded. The novel feat ure of this analysis is that we concentrate on the measure on the space of paths. In particular, under the assumption that the set of unbounded paths has positive measure, we are able to establish a useful ergodic property of the measure. We use this to prove results about the number and geometry of infinite light beams. Extensions to higher dimensions are considered.