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Glauber dynamics on trees and hyperbolic graphs
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  • Published: 27 December 2004

Glauber dynamics on trees and hyperbolic graphs

  • Noam Berger1,
  • Claire Kenyon2,
  • Elchanan Mossel3 &
  • …
  • Yuval Peres4 

Probability Theory and Related Fields volume 131, pages 311–340 (2005)Cite this article

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Abstract.

We study continuous time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap |λ1−λ2|) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in n. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that for general graphs, if the relaxation time τ2 satisfies τ2=O(1), then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp.

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Author information

Authors and Affiliations

  1. University of California, Berkeley, USA

    Noam Berger

  2. LRI, UMR CNRS, Université Paris-Sud, France

    Claire Kenyon

  3. University of California, Berkeley, USA

    Elchanan Mossel

  4. University of California, Berkeley, USA

    Yuval Peres

Authors
  1. Noam Berger
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  2. Claire Kenyon
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  3. Elchanan Mossel
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  4. Yuval Peres
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Corresponding author

Correspondence to Noam Berger.

Additional information

Research supported by Microsoft graduate fellowship.

Supported by a visiting position at INRIA and a PostDoc at Microsoft research.

Research supported by NSF Grants DMS-0104073, CCR-0121555 and a Miller Professorship at UC Berkeley.

Acknowledgement We are grateful to David Aldous, David Levin, Laurent Saloff-Coste and Peter Winkler for useful discussions. We thank Dror Weitz for helpful comments on [19].

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Cite this article

Berger, N., Kenyon, C., Mossel, E. et al. Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Relat. Fields 131, 311–340 (2005). https://doi.org/10.1007/s00440-004-0369-4

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  • Received: 01 September 2003

  • Revised: 16 March 2004

  • Published: 27 December 2004

  • Issue Date: March 2005

  • DOI: https://doi.org/10.1007/s00440-004-0369-4

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Keywords

  • Relaxation Time
  • Mutual Information
  • Mathematical Biology
  • Continuous Time
  • Potts Model
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