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Nontrivial Phase Transition in a Continuum Mirror Model

Authors:	Matthew Harris

Institution:	(1) Department of Technical Mathematics and Informatics, Delft University of Technology, Mekelweg 4, 2628, CD Delft, The Netherlands;(2) Present address: Philips Research Labs, Weisshausstrasse 2, 52066 Aachen, Germany

Abstract:	We consider a Poisson point process on $\mathbb{R}^2$ with intensity , 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 _H^ $ with 0 < $\lambda _H^$ < for which, if $\lambda$ < $\lambda _H^$ , light leaving the origin in all but a countable number of directions will travel arbitrariliy far from the origin with positive probability. Also, if $\lambda$ > $\lambda _H^$ , light from the origin will almost surely remain in a bounded region.

Keywords:	percolation wind tree model Lorenz model phase transition
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