Infinite paths in a Lorentz lattice gas model

Abstract

We consider infinite paths in an illumination problem on the lattice ℤ2, where at each vertex, there is either a two-sided mirror (with probability p ≥ 0) or no mirror (with probability 1 - p). The mirrors are independently oriented NE-SW or NW-SE with equal probability. We consider 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 feature 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.