, Volume 131, Issue 3, pp 311-340
Date: 27 Dec 2004

Glauber dynamics on trees and hyperbolic graphs

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 |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.

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].