Nontrivial Phase Transition in a Continuum Mirror Model

We consider a Poisson point process on R with intensity (lambda), and at each Poisson point we place a two sided mirror of random length and orientation. The length and orientation of a mirror is taken from a fixed distribution, and is independent of the lengths and orientations of the other mirrors. We ask if light shone from the origin will remain in a bounded region. We find that there exists a (lambda)with 0 < (lambda)< (infinity)for which, if (lambda)< (lambda), light leaving the origin in all but a countable number of directions will travel arbitrariliy far from the origin with positive probability. Also, if (lambda)> (lambda), light from the origin will almost surely remain in a bounded region.