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 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.Comment: To appear in Probability Theory and Related Field