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The spectrum of the random environment and localization of noise
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  • Published: 19 May 2009

The spectrum of the random environment and localization of noise

  • Dimitris Cheliotis1 &
  • Bálint Virág2 

Probability Theory and Related Fields volume 148, pages 141–158 (2010)Cite this article

  • 317 Accesses

  • 1 Citations

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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.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Authors and Affiliations

  1. EURANDOM, Eindhoven University of Technology, L.G. 1.26, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Dimitris Cheliotis

  2. University of Toronto, 40 St George St., Toronto, ON, M5S 2E4, Canada

    Bálint Virág

Authors
  1. Dimitris Cheliotis
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  2. Bálint Virág
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Correspondence to Dimitris Cheliotis.

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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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Cite this article

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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  • Received: 30 April 2008

  • Revised: 10 April 2009

  • Published: 19 May 2009

  • Issue Date: September 2010

  • DOI: https://doi.org/10.1007/s00440-009-0225-7

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Mathematics Subject Classification (2000)

  • 15A52
  • 05C50
  • 60K37
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