Probability Theory and Related Fields

, Volume 131, Issue 3, pp 311–340

Glauber dynamics on trees and hyperbolic graphs

  • Noam Berger
  • Claire Kenyon
  • Elchanan Mossel
  • Yuval Peres

DOI: 10.1007/s00440-004-0369-4

Cite this article as:
Berger, N., Kenyon, C., Mossel, E. et al. Probab. Theory Relat. Fields (2005) 131: 311. doi:10.1007/s00440-004-0369-4


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 |λ1λ2|) 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Noam Berger
    • 1
  • Claire Kenyon
    • 2
  • Elchanan Mossel
    • 3
  • Yuval Peres
    • 4
  1. 1.University of CaliforniaBerkeleyUSA
  2. 2.LRI, UMR CNRSUniversité Paris-SudFrance
  3. 3.University of CaliforniaBerkeleyUSA
  4. 4.University of CaliforniaBerkeleyUSA