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

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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.