Abstract
We consider random walk on a mildly random environment on finite transitive d-regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized.
The graphs of the noise covariance structure for d = 4, 3, 2.1 from above.
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Acknowledgments
This research is supported by the Sloan and Connaught grants, the NSERC discovery grant program, and the Canada Research Chair program (Virág). We thank Amir Dembo for encouraging discussions.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Cheliotis, D., Virág, B. The spectrum of the random environment and localization of noise. Probab. Theory Relat. Fields 148, 141–158 (2010). https://doi.org/10.1007/s00440-009-0225-7
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DOI: https://doi.org/10.1007/s00440-009-0225-7