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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Deterministic approximations of random reflectors


Authors: Omer Angel, Krzysztof Burdzy and Scott Sheffield
Journal: Trans. Amer. Math. Soc. 365 (2013), 6367-6383
MSC (2010): Primary 37D50
Published electronically: June 3, 2013
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Abstract: Within classical optics, one may add microscopic ``roughness'' to a macroscopically flat mirror so that parallel rays of a given angle are reflected at different outgoing angles. Taking the limit (as the roughness becomes increasingly microscopic) one obtains a flat surface that reflects randomly, i.e., the transition from incoming to outgoing ray is described by a probability kernel (whose form depends on the nature of the microscopic roughness).

We consider two-dimensional optics (a.k.a. billiards) and show that every random reflector on a line that satisfies a necessary measure-preservation condition (well established in the theory of billiards) can be approximated by deterministic reflectors in this way.


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Additional Information

Omer Angel
Affiliation: Department of Mathematics, University of British Columbia, 121 - 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2
Email: angel@math.ubc.ca

Krzysztof Burdzy
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: burdzy@math.washington.edu

Scott Sheffield
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 2-180, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: sheffield@math.mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05851-5
PII: S 0002-9947(2013)05851-5
Received by editor(s): March 4, 2012
Received by editor(s) in revised form: April 7, 2012, and April 9, 2012
Published electronically: June 3, 2013
Additional Notes: The first author’s research was supported in part by NSERC and by the Sloan Foundation. The second author’s research was supported in part by NSF Grant DMS-0906743 and by grant N N201 397137, MNiSW, Poland. The third author’s research was supported in part by NSF Grant DMS 0645585.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.