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Probability Theory and Related Fields
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.
The final publication is available at Springer via http://dx.doi.org/10.1007/s00440-004-0369-4.
Berger, N., Kenyon, C., Mossel, E., & Peres, Y. (2005). Glauber Dynamics on Trees and Hyperbolic Graphs. Probability Theory and Related Fields, 131 (3), 311-340. http://dx.doi.org/10.1007/s00440-004-0369-4
Date Posted: 27 November 2017
This document has been peer reviewed.