Probability Theory and Related Fields

, Volume 131, Issue 3, pp 311-340

First online:

Glauber dynamics on trees and hyperbolic graphs

  • Noam BergerAffiliated withUniversity of California Email author 
  • , Claire KenyonAffiliated withLRI, UMR CNRS, Université Paris-Sud
  • , Elchanan MosselAffiliated withUniversity of California
  • , Yuval PeresAffiliated withUniversity of California

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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 |λ 1λ 2|) 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 τ 2 satisfies τ 2=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.