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Random Reflections in a High-Dimensional Tube

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Abstract

We consider light ray reflections in a d-dimensional semi-infinite tube, for \(d\ge 3\), made of Lambertian material. The source of light is placed far away from the exit, and the light ray is assumed to reflect so that the distribution of the direction of reflected light ray has the density proportional to the cosine of the angle with the normal vector. We present new results on the exit distribution from the tube, and generalizations of some theorems from an earlier article, where the dimension was limited to the cases \(d=2\) and \(d=3\).

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Acknowledgments

The authors would like to thank Sara Billey for very helpful advice. The second author is grateful to Microsoft Corporation for the allowance on Azure where the simulation illustrated in Fig. 6 was performed. We are grateful to the anonymous referee for many suggestions for improvement.

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Correspondence to Tvrtko Tadić.

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KB: Research supported in part by NSF Grant DMS-1206276. TT: Research supported in part by Croatian Science Foundation Grant 3526.

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Burdzy, K., Tadić, T. Random Reflections in a High-Dimensional Tube. J Theor Probab 31, 466–493 (2018). https://doi.org/10.1007/s10959-016-0703-7

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  • DOI: https://doi.org/10.1007/s10959-016-0703-7

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